On the interaction between small- and large-scale convection and postglacial rebound flow in a power-law mantle

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Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

[41

On the interaction between small- and large-scale convection and postglacial rebound flow in a power-law mantle

H a r r o S c h m e l i n g *

Department of Mineralogy and Petrology, Institute of Geology, Uppsala University, Box 555, S-751 22 Uppsala (Sweden)

Received November 5, 1986; revised version received April 9, 1987

Some consequences arising from the superposition of flows of two different kinds or scales in a non-Newtonian mantle are discussed and applied to the cases mantle convection plus postglacial rebound flow as well as small- plus large-scale mantle convection, If the two flow types have similar magnitude, the apparent rheology of both flows becomes anisotropic and the apparent viscosity for one flow depends on the geometry of the other. If one flow has a magnitude significantly larger than the other, the apparent viscosity for the weak flow is linear but develops direction-dependent variations about a factor n (n being the power exponent of the rheology). For the rebound flow lateral variations of the apparent viscosity about at least 3 are predicted and changes in the flow geometry and relaxation time are possible. On the other hand, rebound flow may weaken the apparent viscosity for convection.

Secondary convection under moving plates may be influenced by the apparent anisotropic rheology. Other mechanisms leading to viscous anisotropy during shearing may increase this effect. A linear stability analysis for the onset of convection with anisotropic linear theology shows that the critical Rayleigh number decreases and the aspect ratio of the movement cells increases for decreasing horizontal shear viscosity (normal viscosity held constant). Applied to the mantle, this model weakens the preference of convection rolls along the direction of plate motion. Under slowly moving plates, rolls perpendicular to the plate motion seem to have a slight preference. These results could be useful for resolving the question of Newtonian versus non-Newtonian or isotropic versus anisotropic mantle rheology.

1. Introduction

M o v i n g l i t h o s p h e r i c p l a t e s c a n b e r e g a r d e d as p a r t o f a l a r g e - s c a l e c o n v e c t i v e m a n t l e flow. I n a d d i t i o n , t h e r e m a y exist s m a l l e r - s c a l e flows i n t h e m a n t l e like s e c o n d a r y , s u b l i t h o s p h e r i c c o n v e c t i o n o r p o s t g l a c i a l r e b o u n d flows. T h e a i m of this p a p e r is to e x p l o r e s o m e c o n s e q u e n c e s of s u c h m u l t i p l e scale flows for t h e case of a n o n - N e w t o - n i a n m a n t l e (see Fig. 1 for a n e x e m p l a r y i l l u s t r a - tion).

D o e s t h e m a n t l e b e h a v e as a N e w t o n i a n or a n o n - N e w t o n i a n f l u i d o v e r l o n g p e r i o d s ? E x p e r i - m e n t a l c r e e p d a t a o n m a n t l e r o c k s h a v e b e e n e x t r a p o l a t e d to m a n t l e c o n d i t i o n s b y n u m e r o u s w o r k e r s (e.g. [1-3]). T h e y all agree t h a t , a b o v e a c e r t a i n t r a n s i t i o n stress, m a n t l e r h e o l o g y will b e d o m i n a t e d b y d i s l o c a t i o n c r e e p e x h i b i t i n g a n o n -

* Present address: Geophysikalisches Institut, Universit~it Karlsruhe, Hertzstrasse 16, D-7500 Karlsruhe, F.R.G.

l i n e a r s t r e s s - s t r a i n rate r e l a t i o n s h i p . T h i s t r a n s i - t i o n stress as well as t h e ( d e v i a t o r i c ) stresses d u e to m a n t l e c o n v e c t i o n h a v e b e e n e s t i m a t e d to b e b o t h of t h e o r d e r of 1 M P a [3-5]. H o w e v e r , a c l e a r d e c i s i o n o n w h e t h e r the m a n t l e is N e w t o n i a n o r n o n - N e w t o n i a n is n o t y e t p o s s i b l e , b e c a u s e the t r a n s i t i o n stress m a y b e u n c e r t a i n b y o n e or t w o o r d e r s of m a g n i t u d e .

M o r e d i r e c t d e t e r m i n a t i o n s of t h e m a n t l e r h e o l - o g y h a v e b e e n c a r r i e d o u t u s i n g d a t a f r o m p o s t - g l a c i a l r e b o u n d . D e s p i t e s o m e a t t e m p t s to e x p l a i n t h e o b s e r v a t i o n s b y a n o n - l i n e a r r h e o l o g y [ 6 - 8 ] m o s t i n v e r s i o n s h a v e a s s u m e d a N e w t o n i a n m a n - tle (see, e.g., [9,10]). H o w e v e r , r e b o u n d a n a l y s i s is n o t v e r y s u i t a b l e for d i s t i n g u i s h i n g b e t w e e n l i n e a r a n d n o n - l i n e a r m a n t l e r h e o l o g y ( [ 1 1 - 1 3 ] , a n d see b e l o w ) .

T h e p o s s i b l e i m p o r t a n c e o f n o n - N e w t o n i a n r h e o l o g y o n m a n t l e c o n v e c t i o n h a s b e e n p o i n t e d o u t b y S c h m e l i n g a n d J a c o b y [14], C s e r e p e s [15], a n d C h r i s t e n s e n [5,16]. T h e s e s t u d i e s c o n c l u d e d , t h a t t h e p l a t e - l i k e b e h a v i o u r of t h e u p p e r b o u n d a r y

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+

Ice

Postgtaciol r e b o u n d

Fig. 1. Illustration of the superposition of a convecting mantle flow and a postglacial rebound flow. The question of whether whole mantle or two-layer convection occurs is not important for the principal approach in this study.

of a convecting earth necessitates a non-Newto- nian lithosphere but could not distinguish whether the mantle below is Newtonian or non-Newto- nian.

Superposition of another flow on a non-Newto- nian convecting mantle is non-linear. One implica- tion was already noted by Weertman [11], Melosh [12], and Turcotte and Schubert [13, p. 325-326].

The superimposed flow appears to obey a linear rheology, if the additional stresses are small compared to the convective stress. Further consequences will be discussed below: if the superimposed flow has the same magnitude as the p r i m a r y flow, the effective viscosity for the p r i m a r y convective flow will be decreased or increased; if the superimposed flow is weaker than the p r i m a r y flow, the apparent rheology for the superimposed flow will be linear but anisotropic; due to laterally varying convective stresses such anisotropic viscosity will show lateral variations. These points could be of importance for problems like postglacial rebound or the onset of secondary small-scale convection beneath a moving lithosphere.

In the following section the constitutive law for multiple scale, non-Newtonian flows will be derived. Then, the implications are qualitatively discussed for post-glacial rebound flow. F o r the onset of small-scale convection a linear stability analysis will be presented, which is valid for the apparent constitutive law in section 2 as well as for viscous anisotropy of different origin.

2. Superposition of stress fields in a power law material

We assume a non-Newtonian constitutive behaviour in the f o r m of a power law (Ostwald-de-

Waele fluid, see, e.g., [17]):

= AT,q-1,j (a)

where eij ^{i s}the strain rate tensor, A is a constant, n is the power, Tij is the deviatoric (viscous) stress tensor, and ~'u is the second stress invariant with the form:

r,I = ½ E ~'ij (2)

i,j

We define the effective viscosity as:

r~j _ ½A-ar~,-" (3)

.('r,j)

- 2ei~ '

We assume a p r i m a r y flow, denoted b y °, which has a constitutive law, in the absence of the superimposed secondary flow of:

• o - - o n - - I o 1 ^o

eij = / t 3 " 1 1 "Tij = 270

¢ij

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Superposing a secondary flow (denoted by ' ) as a result of the additional stress tensor ~'ij the total t

strain rate is:

~ij = A ~ - ~ ( ~ 3 + ~,~) (5)

with:

The effective viscosity for b o t h the p r i m a r y and secondary flows m a y have changed in any combination of the following three cases:

o t o

(l) "r~j and T~j are close to parallel so that ~-~j and ~'~} have non-zero contributions to the same c o m p o n e n t s with the same sign. Then ~'i~ will be

° If ° = ~'ij, the effective viscosity t

larger than ~'i~. ~'~j

for both the p r i m a r y and secondary flow will be

(3)

reduced by a factor 2 ~-" compared to the flows without superposition. This can be seen by insert-

t o

ing r,~ + r u = 2 r u into (2) and (2) into (3).

o /

(2) rgj and r,j are close to orthogonal so that the scalar product Y~,,jz~r,~ ---- 0. Then the effective viscosity will still be reduced; for ri~ = ril by as much as a factor 2 ° - " ) / 2 . This can be seen by inserting the non-zero ~ components and the non-zero ~'kl components (where (i, j ) ~ (k, l)) into (2). From rl~ = rn follows ^{t h a t} E i , j T i ; 2 = ~ k l ' r ; } , and thus the total r n = 21/2r~. With (3) the total

= 2 ° - ' ) / 2 ~ ° where ~o is the viscosity of the primary flow.

(3) If, however, riy and ri~ superimpose antiparallel (riy and r,j have the same non-zero com- i

ponents, but with different sign), the effective viscosity will i n c r e a s e significantly. In the special case of r~j ! just cancelling r,~, the effective viscosity will theoretically become infinite.

Examples for superpositions of "parallel, orthogonal, and antiparallel" stress tensors in two dimensions are given in Table 1. The different effective viscosities for different r/j components may be regarded as an apparent rheological anisotropy.

For the particular case of one of the flows having a small amplitude compared to the other, the rheology "seen" by the weak flow appears anisotropic. Assume that the magnitude of the secondary flow is small compared to the primary flow, i.e.:

• ' O

11 rij II << il "fij II (7)

Neglecting all squares of ri~, (6) gives the second

TABLE 1

Examples for two-dimensional stress tensors, their scalar prod- ucts, and effective viscosity in "parallel, orthogonal, and antiparallel" superposition, a, b > 0

P a r a l l e l O r t h o g o n a l A n t i p a r a U e l

,,; [o oo] [o °a] [o °a]

,, o ] [o [o' o]

o /

Zi,j~ij Tij > 0 0 < 0

~/~0 <1 <1 or =1 >1

invariant of the two superimposed flows:

f

¹ ^{0 2}

z )1':

^o ^z

r i i = ~ ~j + 2 ~jr~ i (8)

k " i , j t , j - j

Expanding (8) into a power series of the second sum and neglecting higher orders than one we obtain:

o 1 ~ o !

q'II • TII + ~ ~-# Tij 7ij (9)

~ ' I I i , j

Before inserting (9) into (5) we evaluate ~'~-1 into a binominal series and neglect orders of rij higher than one:

oo,[. :>,>:-1

Inserting (10) into (5) gives, again neglecting squares of 5}:

. . . . - 1 ^o+ ^_ 1) e i j -- A T I I 'Tij

Z[To

I i j T i j ) t'~

i , j

X o --I- A o n - - 1 ,

2q~2 ~ij - s : ' I I "rij (11) In (11) we can identify the first term on the right-hand side with ei~ (see equation (4)). The remaining strain rate can be identified with the secondary strain rate after superposition, ~ :

Z ~ t i j T i j )

.t ~ A on--1 1) i,j ^o ^,

eij ..'t n n - 0 2 riJ + rij (12)

2 ' r i i

In fact, the first term in brackets of equation (12) may also be regarded as a change of the primary flow due to superposition of the secondary flow.

However, as long as one is interested in the total effect of a secondary small stress r/s , it is most appropriate to describe all resulting deviations from the initial primary strain rate ~ by a secondary strain rate ~:j according to (12).

Inspection of equation (12) shows the possibility of anisotropic b e h a v i o u r : if a particular component rgj acts in a direction for which ziy = 0 (i.e., I

orthogonal superposition), (12) reduces to:

• , _ - o . - i , ( 1 3 )

e i j - - / I T I I 'Tij

If, on the other hand, another component, say "r~t, acts in a direction for which ~ is important (i.e.,

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antiparallel or parallel superposition), the first term in the brackets of (12) will contribute significantly to the strain rate. Assuming, for example, that Tk~ is the only important contribution to TI~

and k 4: l, (12) reduces to:

. . . -1 , (14)

e k l = n ^"Ag"II "Tkl

The same is true for pure shear where, however, always two important components have to be considered. The constitutive law (14) is linear with

° / ¢

respect to ekt and ~'kt substantiating the introduc- tory statement, that a small additional flow will

"see" a linear rather than a power-law rheology.

In addition, the effective viscosity for the kl-component of the secondary flow field for (anti)parallel superposition is smaller by the factor 1 / n compared to the /j-component for orthogonal superposition (equation (13)). For the secondary flow the material appears not only to be linear but also anisotropic; it is weaker if the stresses act parallel, but stronger for orthogonal addition. This apparent anisotropy has already been noted by Fletcher [18] and Smith [19], during the analyses of the onset of folding in non-Newtonian materials.

As another example, let us superimpose a primary pure shear flow (I-1] = -~'3~ = c, else O) with a triaxial secondary flow 0"33 = - c ' , ~'11 = "r22

= c'/2). Inserting these stresses into (12) we arrive at the anisotropic constitutive law for the secondary flow:

e l l = A l c l " - ' . ~(n - 1 / 3 ) ~'11

O~z = A lc [ n--1 9"22 (15)

e33 ⁼ A lcl "-1" 3( n + 1 / 3 ) ~'3'3

Assuming for example n = 3 the viscosities in 2- and 3-directions are larger than that in 1-direction by a factor of 4 and 2.5, respectively.

3. Discussion

To explore the above effects in terms of possible geodynamical flows, first the stresses associated with mantle convection have to be estimated.

Using results from boundary larger theory for high Rayleigh number convection [13,20], an average stress in a constant viscosity convection cell

can be derived as:

= 17(

~/x )1/3(

pgolmz)2/3Rac 2/3 (16) where Ra is the Rayleigh number ( = p g a A T h 3 / t~l), Rac is the critical Rayleigh number above which convection is possible, and is usually of the order 10 3, t¢ is the thermal diffusivity, g is the gravity acceleration, a is the thermal coefficient of expansion, and AT is the temperature difference between top and bottom of the convecting layer.

With likely values for the mantle (16) gives mean stresses between 0.5 and 10 MPa. The stresses derived from the boundary layer analysis (16) are also in good agreement with finite difference calculations of mantle convection (Schmeling, unpublished results). Convection models with a high viscosity lid or with temperature and pressure dependent viscosity show that the stresses may increase up to several 10 MPa in either the lid or the highly viscous downwelling regions [5,21].

3.1. Postglacial rebound

The consequences of a possible superposition of postglacial rebound flow on non-Newtonian mantle convection will be examined qualitatively.

Analyses of postglacial rebound (e.g., [10,13]) al- low estimates of deviatoric stresses of several MPa during the early stages of uplift which may have decreased down to the order of 1 MPa at present.

Thus, situations may exist in which rebound stresses are either small or similar compared to convective stresses.

If rebound flow takes place in a convecting mantle, stress regimes of "parallel, orthogonal or

Rebound Convection

Fig. 2. Schematic illustrations of the main stress regimes in rebound and convective flow. Assuming two-dimensionality, the regions are characterized by horizontal pure shear (ps, ['rll I,I ~221 >> I'rl21) or horizontal simple shear (ss, [ "r12 [ >>

I zn I, I z221).

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antiparallel" superposition might occur (see Fig.

2). If the flows have similar magnitude, it follows from the discussion in section 2 that the effective viscosities in the different regimes will be different. If the rebound flow is weak compared to convection, the rebound flow will "see" a linear but anisotropic rheology with apparent viscosities of ~ ° / n for parallel or antiparallel superposition and ~/o for orthogonal superposition (7/° was the effective viscosity for convection). The mantle rheology appears to be heterogeneous depending on the convective flow pattern. Taking for example an n-value of 3, lateral variations of the apparent rebound viscosity by a factor of similar magnitude (3) are possible. Viscosity inversions do not exclude such variations (Peltier, personal communication). It should, however, be noted that non-homogeneous viscous anisotropy of different origin may increase the apparent heterogeneity too (see below). A temperature-dependent viscosity in combination with laterally varying tempera- tures in a convecting mantle might lead to lateral variations of viscosity of significant magnitude (these variations are, however, reduced in a non- Newtonian mantle [5]).

The apparent rheological anisotropy of a rebound flow superimposed on non-Newtonian convection would have important consequences for the geometry and relaxation time of the rebound flow itself. Christensen ([22]; and unpublished manuscript, 1987) has shown that the relaxation time for two-dimensional rebound flow in a half space with constant but anisotropic viscosity is proportional to the geometric mean of the simple and pure shear viscosity components, and that the flow penetrates deeper than in the isotropic case.

If these results are applied to the apparent anisotropy as expressed in equations (13) and (14), the postglacial rebound analysis would lead to an apparent viscosity of ~ ° / v ~ , thereby sampling deeper parts of the mantle compared to the New- tonian case.

3.2 Onset of small-scale convection

The results of section 2 may have some consequences for the onset of second scale convection beneath moving plates. The deviation of observed heat flow data from a simple lithospheric cooling model for oceanic lithosphere older than 70 Ma was the motivation for Richter [23], Richter and

Fig. 3. Illustration of the large-scale shear flow associated with a moving plate and sublithospheric second scale convection rolls, either normal (N) or parallel (P) to the moving plate. The depth of the large-scale return flow is not crucial for the present study.

Parsons [24] and others to consider the possibility of small-scale convection beneath lithospheric plates which are moving as a consequence of large-scale convection. Richter [23] and Richter and Parsons [24] have shown that convection rolls aligned along the direction of shear due to plate movements (denoted here as parallel P-rolls) are preferred compared to rolls normal to plate movements (N-rolls; see Fig. 3). For this reason most subsequent workers have only considered P-rolls [25-28]. In this section I will show that a strongly non-Newtonian or anisotropic mantle beneath moving plates would counteract the preference of P-rolls.

The critical Rayleigh number for the onset of two-dimensional convection in the form of N-rolls in a Newtonian fluid is not affected by a large-scale horizontal shear [23]. However, the amplitude of N-rolls for weakly supercritical Rayleigh numbers is reduced and may drop to zero for increasing shearing [23]. If one now assumes a non-Newto- nian fluid under horizontal shear, the constitutive law valid for the onset of two-dimensional convection in the form of N-rolls can be derived from (12):

1 1 1

N I e l l = 2 ~ n T l l ; e22 = " ~ n T22; el2 = ~ t T 1 2

(17)

where ~n, ~t are the normal or tangential apparent viscosities. In contrast, P-rolls "see" an isotropic rheology with the ~n-Viscosity:

P: eij = - - ¢ i 1 (18)

2~n J

We define the anisotropy factor:

s = ~/./~/t ( - n) (19)

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where the equality with the power n is valid for a power-law rheology. It can already be noted that for s > 1 (n > 1) the N-case appears less viscous than the P-case due to a softer apparent viscosity for the 12-components.

Beneath moving plates the viscous anisotropy factor s might be larger than n due to additional mechanisms. For example, dislocation creep implies that for uniaxial stress the strain rate is proportional to the velocity of the dislocations v (which is proportional to z) and their density ( _ ~ n - l ) [29]. An additional orthogonal stress tensor zij may not be able to activate all pre-exist- p

ing dislocations to move with v' (which would be proportional to

II "ri~ II)

into the new (')-direction.

This would lead to an increase in apparent anisotropy (L. Fleitout, personal communication, 1987). Furthermore, a shear flow might cause viscous anisotropy due to a preferred orientation of crystallographic fabrics or streaks of eclogitic material in a " m a r b l e cake" mantle (U. Christen- sen, unpublished manuscript, 1987).

Assuming free slip and constant temperature conditions at the top and b o t t o m of the convecting layer, a linear stability analysis for an isotropic, constant viscosity (i.e., for P-rolls) can be carried out yielding a critical Rayleigh number (see, e.g., [13, pp. 274-279]):

q.j. 4

P: R a c = ( a ) ( 0 2 + 1) 3 (20) where a is the aspect ratio (width/thickness) of the convection cells. Ra¢ has a minimum for a = a~ = V~-. Taking now the anisotropic constitutive law (17) a similar linear stability analysis yields:

l ( r r ] 4- 2 11

N: R a c = s ~ a ! (a +1)[a4+(4s-2)a2+

(21) This critical Rayleigh number is shown as a function of aspect ratio for different values of s in Fig.

4, where (20) is represented by the curve s = 1.

The right-hand side parts of the Ra¢(a) curves decrease with increasing s, indicating that convection will start at smaller Rayleigh numbers with larger aspect ratios. The inset diagram shows the minimum critical Rayleigh number and the corresponding aspect ratio a c as functions of s. The asymptotic behaviour can be described as a c

---+ vl2s 1/4 and Rac~" --+ 389.6 as s ~ ~ . A quali- tative similar behaviour was already found by Richter and Daly [30] who used a simpler formu- lation of anisotropy.

The above analysis is based on viscous anisotropy caused by a strong primary shear flow, which implies that the secondary flow is assumed to be comparatively small. As Richter [23] and Richter and Parsons [24] found this is just the condition to suppress steady state N-rolls in an isotropic mantle. Nevertheless, strong viscous anisotropy due to shearing could have an important influence in the following two situations (a) and (b). The effects discussed in cases (c) and (d) result from the principal behaviour of non-Newto- nian rheology in multiscale flows.

(a) Suppose that the lithosphere cools sufficiently fast for the Rayleigh number for second scale convection to increase to the stage where steady-state convection overcomes the large-scale shear. It may happen that the growth of small perturbations could not keep pace with the increasing Rayleigh number. Then, in such an over- critical but still not convecting layer, Newtonian isotropic rheology implies that N- and P-rolls are almost equally probable. However, for non-New- tonian or anisotropic rheology the above analysis is applicable to the early stages of convective instability implying a preference of N-rolls. Such N-rolls will be preserved if their growth rates are significantly larger than the strain rate of the primary shear flow.

To test this possibility the growth rate ( = inverse time for e-fold growth) can be derived from the linear stability analysis carried out for the anisotropic rheology [17]:

7 = ~ - ~ - r t 1 + 7 ~--~-a - 1 (22) However, the free-slip and constant temperature boundary conditions are rather unrealistic for sublithospheric convection. For a better estimate of Ra/Ra c one can use the results of Houseman and McKenzie [26], who calculated the critical Rayleigh number of an initially isothermal system consist- ing of a fluid layer covered by a conductive lid (the lithosphere), as it cooled from the top (by downward diffusion of a thermal boundary layer).

If the Rayleigh number is based on the temperature difference, 3T, between the top of the fluid

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1000

Rn c

r

500

0 0

c E rY Z

7 0 0 &

600

500

Z

3 .~

,<

4-00 . . , i . . . . , . . . . , , , I

5 10 15 > S

I I I I

> A s p e c f r a f i o o

Fig. 4. Critical Rayleigh number Ra c as a function of aspect ratio, a, for different degrees of anisotropy, s ( = ~/,/7/t). The inset diagram shows the minimum critical Rayleigh number (left vertical scale) and the corresponding aspect ratio (right vertical scale) as a function of s. Boundary conditions of the linear stability analysis are: free slip and constant temperature at top and bottom.

l a y e r a n d T = 0.999T 0 ( T o b e i n g the initial tem- p e r a t u r e ) , a n d the thickness of this t h e r m a l b o u n d a r y layer, 8, t h e y c a l c u l a t e a critical R a y l e i g h n u m b e r R a c o f 18,600. W e e s t i m a t e R a b y ~ = 400 km, 8 T = 1 0 0 - 4 0 0 K , O = 3500 kg m - 3 , a = 3.7 × 10 - 5 K -1, K -- 10 . 6 m 2 s -1 a n d t e n t a t i v e l y i n s e r t

R a , R a c a n d h = 600 k m i n t o (22). I f the u p p e r m a n t l e viscosity is s m a l l e r t h a n a b o u t 1019-102o P a s the g r o w t h r a t e c a n b e c o n j e c t u r e d to b e l a r g e r or o f the s a m e o r d e r as the s t r a i n r a t e o f the p r i m a r y s h e a r flow (which m a y b e e s t i m a t e d b e - t w e e n 10 -14 a n d 10 -16 s -1 d e p e n d i n g o n the d e p t h of the r e t u r n flow).

Such viscosities c a n n o t b e r u l e d o u t for y o u n g o c e a n i c u p p e r m a n t l e . T h e y w o u l d i m p l y t h a t for

a m a n t l e b e l o w a s p r e a d i n g p l a t e b e c o m i n g su- p e r c r i t i c a l sufficiently fast c o n v e c t i v e rolls n o r m a l to the s p r e a d i n g d i r e c t i o n m a y b e slightly pre- f e r r e d d u r i n g the early stages o f the s e c o n d scale c o n v e c t i o n . It is n o t clear at the m o m e n t if this p r e f e r e n c e is sufficient to c o m p e t e w i t h R i c h t e r ' s p r e f e r e n c e for P-rolls o r o t h e r d i s t u r b a n c e s p a r a l - lel to the s p r e a d i n g d i r e c t i o n such as t r a n s f o r m faults.

(b) If a s t r o n g s h e a r flow a s s o c i a t e d w i t h a m o v i n g p l a t e c h a n g e s its d i r e c t i o n a b r u p t l y by, say, 90 o, P-rolls b e n e a t h this p l a t e will e v e n t u a l l y also c h a n g e their o r i e n t a t i o n . Since c o n d i t i o n (7) a p p l i e s in this case, the slight p r e f e r e n c e of N - r o l l s d u e to a n i s o t r o p i c (or n o n - N e w t o n i a n ) r h e o l o g y

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will retard, but not prevent a reorientation of the convective rolls compared to the isotropic case.

(c) If the magnitude of the large-scale shear flow is of the same order or less than the second scale convection, Richter [23] and Richter and Parsons [24] predict a more or less pronounced preference of P-rolls. For this case condition (7) fails and thus the linear stability analysis leading to (21) is no longer valid. Nevertheless, the implications from equations (5) and (6) as discussed in section 2 apply if the large- and small-scale flows have similar amplitudes. It still follows that secondary N-rolls will experience an anisotropic and, due to spatial variations of the convective stress field, a heterogeneous theology: regions of normal stresses will "see" the same viscosity as P-rolls, while regions of horizontal shear stress will

" s e e " a lower or higher viscosity if the shear has the same or opposite sign as the large-scale shear flow, respectively. It then follows that P-rolls are less preferred than predicted by Richter [23] and Richter and Parsons [24], if they are compared to N-rolls with the same sense of rotation as the large-scale shear flow in a non-Newtonian mantle.

(d) The rheology may control the large-scale flow if the secondary convection is significantly stronger than the large-scale shear flow. For P- rolls, the large-scale flow will "see" a viscosity proportional to TII -n, where "/'ii is determined by the stresses of the secondary convection. However, for N-rolls, the large-scale flow will "see" a viscosity proportional to

1/n. "rIi-"

in regions of horizontal shear of the N-rolls. Thus, on the average the mantle appears softer if lithospheric plates are overriding strongly convecting N-rolls rather than P-rolls.

The question of whether rheological preference of N-rolls predicted above could outpace the kinematic preference of P-rolls assumed so far, cannot be answered here. A thorough analysis of oceanic gravity anomalies is needed which com- pares the wavelength-spectrum parallel to the direction of spreading to the perpendicular spectrum.

4. Conclusions

This paper outlines some consequences which will arise in a non-Newtonian mantle if flows of different kind or scale occur at the same time. The

results might be useful if applied to particular situations like mantle convection + postglacial re- b o u n d flow or large-scale + small-scale convection. It is found that for the superimposed flows the apparent viscosities are anisotropic and, depending on spatial variations of the flow stress field, heterogeneous. For the special case of one flow having a significantly larger magnitude than the other, the apparent anisotropy is quantified showing that the viscosities for different stress components differ by a factor around n (n being the power of power-law rheology).

Applying the results to postglacial rebound flow superimposed on a convecting mantle, it is argued that lateral variations of the apparent rebound viscosity of the order of at least 3 should be expected and that the flow geometry and relaxation time might be different from the Newtonian case. Furthermore, convective flow may be en- hanced or retarded in rebound areas. More work on rebound analysis focusing on lateral variations of the viscosity and on the influence of anisotropy is necessary to elucidate the question of Newto- nian versus non-Newtonian or isotropic versus anisotropic rheology (see, e.g., Sabadini [31] for a rebound analysis in a heterogeneous mantle).

Non-Newtonian a n d / o r anisotropic rheology of the mantle beneath moving plates complicates the question of whether or not the secondary convection forms two-dimensional rolls parallel to the plate motion. A hnear stability analysis for a layer with anisotropic rheology shows that the critical Rayleigh number decreases while the aspect ratio of the convection cells increases with increasing degree of anisotropy. If applied to a non-New- tonian mantle below moving plates, the preference for secondary rolls aligned along with the direction of plate motion [23,24] seems less pronounced. Under slow-moving plates it is conjec- tured that secondary convection rolls perpendicular to the plate motion are slightly preferred. If such rolls could be verified from observational data, this would be an indication for a non-New- tonian or viscously anisotropic mantle.

Acknowledgements

I greatly appreciate the helpful discussions with U. Christensen, W.R. Peltier, and G. Marquart. U.

Christensen and L. Fleitout are thanked for their

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constructive reviews. The idea of this paper was born during a course on rheology I gave at our institute and I want to thank the students of that course for their inspiring discussions. I am indebt- ed to K. Gloersen who kindly processed the words and to C. Wernstrt~m for making the drawings.

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