Schedule • Basic equations you need to know in geodynamics • Fundamentals of fluid dynamics • Vectors and tensors • Conservation laws and constitutive law • 1D examples • Dynamic similarity • Fundamentals of mantle convection • Boussinesq approximation • Rayleigh number and Nusselt number • Stability analysis of mantle convection Examples: Post-glacial rebound

Initial: h(t=0) = h0 cos(2π/Lx) Assumption: L >> h0

Coastline photo from Turcotte & Schubert, 2002) Example: Ductile structures

http://www.sjsresource.com.au/images/Photo%20Galleries/Structural%20Geology/Folds/multilayer %20behaviour%20Folding,%20Ticino%203.jpg

O(1) m Example: Ductile structures

http://www.geosci.usyd.edu.au/users/prey/Teaching/Geol-1002/HTML.Lect4/LargeFold.jpg

O(100) m Example: Ductile structures

http://en.wikipedia.org/wiki/Fold_mountain#mediaviewer/File:Zagros_1992.jpg

O(1000) km A convecting Earth Plate tectonics provides a framework to understand global seismicity, volcanism and mountain building processes.

Plate motion and the interaction of these plates underpin this theory

* Earth has a heat source coming from the core! ! * Heat is converted into motion via density variations

* Drifting continents * Spreading centres * Subduction zones * Post glacial rebound * Hotspots * Heat budget Thermal convection http://dreamtigers.wordpress.com/2011/05/11/plate-tectonic-metaphor-illustrations-cmu/

Variations in temperature cause small changes in the fluid density

Buoyancy forces cause cold http://asapscience.tumblr.com/post/50419005208/the-earths-center-is-out-of-sync-we-all-know (compressed, i.e. higher density) material to sink http://www.see.leeds.ac.uk/structure/dynamicearth/convection/models.htm Early experiments

Brittle material

Convection = flow driven by internal forces (buoyancy)

In the Earth, density variations are attributed to pressure, Viscous fluid “convection” temperature and composition

Convection drives large-scale dynamics within in the Earth’s mantle Modern experiments Fundamentals of fluid dynamics Frames of reference DT D =  2T Material derivative Dt r Dt

Stationary frame (Eulerian) convective rate of change due to transport DT @T @T := + v of material to a different position (x) with Dt @t x @x respect to a fixed coordinate system local rate of change due to DT @T = + v T temporal variations at a Dt @t ·r position x Moving frame (Lagrangian)

DT @T coordinate@T system is transported with the := + v material, thus only the local rate of change Dt @t x @x remains Conservation of mass • General continuity equation: @⇢ ! + (⇢v)=0 ! @t r·

• Incompressible fluid For many geological material (such as the Earth’s crust and mantle), over long time-spans, one may assume an incompressible condition (e.g. the density of each material point does not change with time) D⇢ =0 v =0 Dt r· Valid assumption when pressure and temperature changes are not very large and no phase transformations (e.g. no large volume changes) occur within the medium Strain

original shape Tensile strain - or - direct strain

✏xx,✏yy deformed shape

Shear strain ✏xy

y Volumetric strain (dilatation)

✏xx + ✏yy x Tensor: Physical meaning ⌧ij ⌧yy ⌧xy direction face normal ⌧yx

⌧xx y

x Strain-rate tensor

• Time rate of change of the strain tensor

! ✏˙xx, ✏˙yy, ✏˙xy [1/s] x! =(x, y) position !v =(vx,vy) velocity

! @vx @vy 1 @vx @vy ✏˙ = ✏˙yy = ✏˙xy = + xx @x @y 2 @y @x ! ✓ ◆ 1 @v @v ✏˙ = i + j ! ij 2 @x @x ✓ j i ◆ • A deviatoric tensor has a mean (average) of zero, e.g.

2 1 1 1 (˙✏xx +˙✏yy)=0 ✏˙kk =0 ✏˙kk =0 2 2 ! 2 kX=1 Stress tensor

=stress⌧ [Pa]p ij ij ij

ij = ⌧ij pijpressure [Pa] 1 p = = ( + + ), for k =1, 2, 3 kk 3 xx yy zz

ij = ⌧ij deviatoricpij stress [Pa] = ⌧ p ij ij ij stress = deviatoric stress - pressure Constitutive law for a fluid • Viscous stress deviatoric! ! ⌧ =2⌘ ✏˜˙ ij ij strain rate [1/s] !deviatoric! stress [Pa] viscosity [Pa s] ! • Incompressible

! ⌧ij =2⌘ ✏˙ij strain rate

• Expanded form (2D) @vx ⌧ =2⌘ ✏˙ =2⌘ xx xx @x @v ⌧ =2⌘ ✏˙ =2⌘ z zz zz @z @v @v ⌧ =2⌘ ✏˙ = ⌘ x + z xz xz @z @x ✓ ◆ Conservation of momentum @v @v @ • General equation: ⇢ i + v i = ij ⇢g @t k @x @x i ✓ k ◆ j ! Dimensional considerations:

! - plate velocity ~ 1 cm/year, 19 ⇢@v/@t 10 ⇠ ! changes occur on 1 Myr timescale 4 - density ~ 3000 kg/m3, g = 10 m/s2 ⇢g 10 ! ⇠

• Mom. balance for a creeping fluid: @ij 0= ⇢gi ! “Stokes flow” @xj

• Primitive equations (v,p) obtained by • inserting relationship for deviatoric stress and pressure • inserting constitutive law • replacing strain-rates by velocity gradients 1D shear flow (Couette)

z

x With:

• Ignore body forces Vx = 5 cm/year 0 • Steady state Vx H ~100 km xz = ⌘ • Constant viscosity h η ~ 1022 Pa s • 1D velocity field σxz ~ 150 MPa • Pressure gradient is zero 1D pipe flow (Poiseuille)

• Ignore body forces

• Steady state

• 1D velocity field: Vz(x)

• Pressure gradient is constant

• Viscosity is constant Dynamic similarity

• Perform an infinite number of experiments to understand the dynamics • or… find a more compact representation of the equations • Non-dimensional numbers BOUNDARY LAYER FLOW AND FLOWFlow IN PIPES AND DUCTSregimes cHAP.5l 101

(a)Re

(b) Re :

(d) Re: l00O Fig. $6. Flow overa cylinderat differentReynolds numbers. flow may seem strange but is a very important phenomenon. Such periodic shedding gives rise to a periodic force on the object, and if coupling occurs with the mechanical system of the object itself a self-sustainedoscillation may result. Resonanceconditions may causecatastrophic effects. The actual position of separation is diffrcult to compute analytically becauseof the interaction of the wake and potential flow region. The wake itself changesthe potential flow pattern upstream and the corresponding pressure gradient along the surface.The separation point depends on the pressure gra- dient along the surface and the turbulence level in the boundary layer. The location of the separation point moves toward the rear (or trailing) edge as the turbulence level increases.The level of turbulence in the boundary layer is affected by the roughness of the surface and the level of turbulence in the free stream outside the boundary layer. Later we will discussthe effect of separation and the boundary layer on drag. In many flows one finds that the boundary layer is so small over the entire body that a solution neglecting viscosity completely (i.e. a potential flow solution) gives accurate results for the pressuredistribution. Such is the case for flow over streamlined airfoils, for example. However, if the body is blunt in the rear and the wake becomes appreciable becauseof boundary layer thickening or separation,the potential flow solution is incorrect except over the front portion of the body where the boundary is thin.

5.2 EXTERNAL FLOWS--SOUNDARY LAYERS

Flow Over a Flat Plate

The flow over a flat plate is characteristic of external flows in general.This flow is shown in Fig. 5-1. The boundary layer thickness is zero at the leading edge and increaseswith distance along the plate surface.The early portion of the boundary layer is laminar but may be followed by a transition region Flow regimes - Issue of scale

Karman vortex sheet caused by wind flowing around the Juan Fernandez Islands of the Chilean coast

O(cm) O(km) Permits meaningful models to be developed

Kincaid (2013) Schellart (2008) Reynolds number of the mantle

xi ui U 1 x0 = u0 = t0 = tp0 = p i h i U h ⇢U 2 ⇢hU Re = ratio of inertial forces to viscous forces ⌘

⇢Du /Dt Du0 /Dt0 | u | = Re | i | ⌘@2u /@x @x @2u /@x @x | i k k| | i0 k0 k0 | CharacteristicMantle values: values for the mantle ρ ∼ ? kg/m3 U ~ ? m/s h ~ ? km η ~ 10? Pas Re ~ ? Fundamentals of mantle convection (buoyancy driven incompressible Stokes flow) Boussinesq approximation • Thermal variations in a fluid lead to small amounts of expansion / contraction. • Expansion results in lowering of density, e.g. resulting in a buoyancy force —> leading to fluid motion

perturbation ⇢ = ⇢0 + ⇢0 ⇢0 ⇢ ⌧ 0 Reference density

⇢0 = ⇢ ↵ (T T ) 0 v 0 Reference temperature corresponding to ref. density

coefficient of thermal expansivity [1/K] Mantle convection eqns.

Compact form:

p + ⌘ v + vT = ⇢ (1 ↵(T T )) geˆ r r· r r 0 0 z v =0 r· DT ⇢C = (k T )+⇢Q p Dt r· r

Constant viscosity and constant conductivity:

p + ⌘ 2v = ⇢ (1 ↵(T T )) geˆ r r 0 0 z v =0 r· DT Q =  2T + Dt r Cp Non-dimensional form (3) Scaling:

(1) Dimensional form: x0 = x/h p + ⌘ 2v = ⇢ (1 ↵(T T )) geˆ 2 r r 0 0 z t0 = t/(h /) v =0 r· T = T/T DT 0 =  2T Dt r v0 = v/(/h) 2 p0 = p/(⌘/h )

(2) Perturbation from (4) Non-dimensional form: 2 background state: 0p0 + 0 v0 = RaT 0eˆ r r z p + ⌘ 2v = ⇢ ↵T geˆ 0 v0 =0 r r 0 z r · DT 0 2 v =0 = 0 T 0 r· Dt0 r DT 2 3 =  T ↵⇢0gTh Ra = Dt r ⌘ Rayleigh number Rayleigh number (Ra) • Describes the relationship between buoyancy and viscosity within a fluid • The Rayleigh number gives a sense of the “vigour” of convection • For small temperature difference (small Ra): conduction is the dominant mode of heat transfer • For larger Ra: convection becomes the dominant mode • The transition between conduction and convection is referred to as the “critical Rayleigh number (Racr)” • Where does this transition occur? Stability analysis

Model definition:

Procedure:

T = Tconductive + T 0 = T0 + T1 2⇡x T (x, y)=f(y)sin 1 ✓ ◆ * For small velocity and T1, we can linearise the system * Energy equation expressed in terms of T1 and Vz 657.5 * “Small” terms are neglected * Solve via separation of variables

3 1 4⇡2b2 4⇡2b2 Ra = ⇡2 + Below Ra critical, ! cr 2 2 ✓ ◆ ✓ ◆ no convection occurs Critical Rayleigh number • More than one definition exists… Numerical experiments

2 0p0 + 0 v0 = RaT 0eˆ r r z 0 v0 =0 r · DT 0 2 = 0 T 0 Dt0 r

↵⇢ gTh3 Ra = 0 ⌘ Numerical experiments Nusselt number (Nu)

• The Nusselt number (Nu) is a ratio of convective heat transfer to conductive heat transfer

• Nu is a non-dimensional number

• In the context of mantle dynamics, it’s defined using horizontally averaged heat fluxes over the upper/lower boundaries z =! H

! L qsurf. 1 @T T ! Nu = = dx qcond. L 0 @z z=H H ! Z ✓ ◆ (assumes constant conductivity, k) z = 0 ! x = 0 x = L

• Nu provides a measure of the efficiency of heat loss through the surface via advection Nu-Ra scaling laws

Problem dependence (variable viscosity) (iso-viscous: moderate Ra) Nu = cRa c =0.27,=0.3185

(iso-viscous: high Ra)

Fukao et al. (2008) Christensen (1984) https://youtu.be/t1UH9Qjy28I Summary • Introduced the fundamental equations which describe the long-time evolution of the mantle • conservation of momentum for highly viscous fluids • incompressibility • dynamic similarity • Boussinesq approximation • important non-dimensional numbers for mantle convection: Rayleigh number (Ra) and Nusselt number (Nu)