Theoretical Geomagnetism. Lecture 3. Core Dynamics I: Rotating Convection in Spherical Geometry

Theoretical Geomagnetism Lecture 3 Core Dynamics I: Rotating Convection in Spherical Geometry 1

3.0 Ingredients of core dynamics Rotation places constraints on motions. Thermal (and chemical buoyancy) produces convection. Spherical shell geometry. Dynamo generated magnetic field will also influence motions. 2

3.0 Ingredients of core dynamics Tangent cylinder barrier to flow Magnetic Field Small Scale Convection Columns (Modified from Aurnou, 2007) 3

Lecture 3: Core Dynamics I 3.1 Equations governing core dynamics 3.2 Stabilizing influence of rotation 3.3 Buoyancy driven flows (without magnetic field) 3.4 Spherical shell geometry and boundary layers 3.5 Summary 4

3.1.1 The Navier-Stokes equation ρ u t + ρ(u )u = P + ρν 2 u Rate of change of flow momentum w.r.t. time Advection of flow momentum along with fluid Force due to pressure gradient Viscous force on fluid Du (known together as flow inertia Dt i.e. the acceleration of a fluid parcel) 5

3.1.2 Adding rotation: The Coriolis Force ρ u t + ρ(u )u +2ρ(Ω u) = P + ρν 2 u Coriolis force due to rotating reference frame Centrifugal acceleration is added to pressure gradient, so P is effective pressure in the rotating frame. u is now the fluid velocity in the rotating reference frame. 6

3.1.3 The Buoyancy force ρ u t + ρ(u )u +2ρ(Ω u) = P + _ ραt g + ρν 2 u 0 0 0 0 0 Temperature differences produce a change in density and so a buoyancy force. The effects of the buoyancy forces are most easily taken into account using the Boussinesq Approximation: - The fluid is assumed to have constant background density ρ 0 with a background temperature field T0, and perturbations T evolving as: T t +(u )T 0 = κ 2 T - Viscous and Ohmic heating effects neglected. - An additional buoyancy term and transport equation is needed to incorporate compositional convection (left away here for the sake of simplicity) - Only dynamic effect of T is through gravity acting on density perturbations ραt as described in the buoyancy force. g 7

3.1.4 The Lorentz force ρ u t + ρ(u )u +2ρ(Ω u) = P + _ ραt g + ρ(j B)+ρν 2 u 0 0 0 0 0 J B = 1 µ 0 (r B) B = B 2 r 2µ 0 Lorentz force: Force due to magnetic field acting on electric currents in fluid. + 1 µ 0 (B r)b Lorentz force can be written as a sum of terms accounting for magnetic pressure and field line curvature effects. The evolution of the magnetic field is as before determined by the magnetic induction equation, B t = (u B)+η 2 B 8

ρ u 3.1.5 Equations governing core dynamics t + ρ(u )u +2ρ(Ω u) = P + ραt g + ρ(j B)+ρν 2 u 0 0 T t +(u )T 0 = κ 2 T B 0 0 0 Terms representing feedback and coupling between the velocity, magnetic and temperature fields t = (u B)+η 2 B u =0 B =0 + Boundary Flow, Thermal and Magnetic BC on ICB & CMB. _ These are the equations of convection-driven, rotating magnetohydrodynamics under the Boussinesq approx. To get some feeling for the flow patterns and dynamics possible for such a system, we will consider some simple examples... 9

3.2.1 Geostrophic Balance For slow motions (so that inertial terms are negligible), and also neglecting buoyancy driving, viscous effects and the Lorentz force, then we can isolate the influence of rotation on fluid motions: 2(Ω u) = P ρ Geostrophic Balance between the Coriolis force and the pressure gradient. P Ω Flow is perpendicular to pressure gradient and involves circulations around regions of high and low pressure. L u 11

3.2.2 Taylor-Proudman Theorem Assuming the rotation vector is parallel to the z axis so Ω =Ωẑ, Taking the curl of both sides, 2(Ωẑ u) = P ρ 2Ω u z = 1 ( P ρ) ρ2 If density gradients are negligible so that ρ = ρ 0, u z =0 Taylor-Proudman theorem i.e. In rapidly rotating fluids, flow patterns are INVARIANT // rotation axis. (or flow varies only in 2D, in planes perpendicular to rotation axis) 12

3.2.2 Taylor-Proudman Theorem Non-rotating case Obstacle "u "z # 0 Rapidly-rotating flow past caseobstacle (From Brenner, 1999) Taylor Column Obstacle GI Taylor! "u "z = 0 Taylor-Proudman Theorem (From Aurnou, Les Houches Lectures, 2007) 13

3.2.3 Geostrophic motions For incompressible fluids, in order for u x x + u y y =0 u/ z =0 then This requires z-independent flow patterns motions to follow contours of constant depth. Ω e.g. Flow patterns move only zonally at fixed latitude in a spherical shell without changing height H. H Geostrophic Contours 14

3.2.4 Quasi-geostrophic motions Forcing by convection, inhomogeneous boundary conditions or magnetic effects will lead to departures from perfect geostrophy. When columnar disturbances are forced off geostrophic contours the resulting motions are known as quasi-geostrophic. To understand such motions it is instructive to begin with the Navier- Stokes equation for a rotating fluid in the absence of any forcing (e.g. buoyancy or Lorentz) and without viscous dissipation: u +(u )u +2(Ω u) = P t ρ 0 Curling this we obtain an evolution eqn for the vorticity ξ = u : ξ t +(u )ξ (ξ )u 2Ω u z =0 This can be re-written in the form of a frozen flux eqn for ξ +2Ω, D(ξ +2Ω) Dt = [(ξ +2Ω) ]u (1) 15

3.2.4 Quasi-geostrophic motions To progress we adopt a quasi-geostrophic scaling for whereby motions are 2D in the (x,y) plane perpendicular to the rotation axis (i.e. geostrophic) but there is also a weaker flow parallel to the rotation axes, which can vary in this ẑ direction: u = u(x, y) x + v(x, y)ŷ + w(x, y, z)ẑ where w<<u,v The approximately 2D flow can therefore be represented in terms of a stream-function representation in terms of a single scalar: ( ) ψ (u, v, 0) = y, ψ x, 0 Substituting into (1), then to leading order the the vorticity equation is, ẑ u (axial) component of D(ξ z + 2Ω) Dt =(ξ z + 2Ω) w z 16

Integrating over the ẑ direction, this yields an expression depending on the axial flow at the boundaries, H D(ξ z + 2Ω) Dt =(ξ z + 2Ω)[w(H) w(0)] Due to non-penetration at the boundaries, DH Dt =[w(h) w(0)] So, Or, 3.2.4 Quasi-geostrophic motions H D(ξ z + 2Ω) (ξ z + 2Ω) DH Dt Dt =0 ( ) D ξz +2Ω =0 (2) Dt H H u w(h) w(0) (From J. Aubert, Les Houches lecture, 2007) i.e. in the absence of forcing or dissipation, potential vorticity is conserved moving with the fluid. 17

3.3.1 Convection-driven flows: Motivation Is the Earth s core convecting? How do we know/can we investigate this? 19

3.3.2 Convection in a rotating plane layer ~ T = T 0 L ~g T = T 1 The equations describing the physical behavior of this system are: @u @t r u =0 + u ru + 2 ẑ u = rp + r 2 u + T gẑ @T @t + u rt = appler2 T A purely conductive solution is: u = 0, T= T 1 +(T 0 T 1 ) z L,p= g T 1 z +(T 0 T 1 ) z2 2L 20

3.3.2 Convection in a rotating plane layer T = T 0 ~ L ~g T = T 1 The question is now whether this conductive equilibrium state is dynamically stable. Stability is related to the evolution of small (infinitesimal) perturbations of the equilibrium state. If they are exponentially amplified, then the equilibrium is said to be unstable. Thus, we consider perturbations of the form (with " 1 ) u = U + "u 0,p= P + "p 0,T = +"T 0 21

3.3.2 Convection in a rotating plane layer T = T 0 ~ L ~g T = T 1 Neglecting terms in " 2 w.r.t. " and omitting the, we obtain the following (linear) perturbation equations: r u =0 @u @t + 2 ẑ u = rp + r 2 u + gt ẑ @T @t + u r =appler2 T 22

3.3.2 Convection in a rotating plane layer T = T 0 ~ L ~g T = T 1 It s customary to write these equations in non-dimensional form with E = L 2,Ra= g(t 1 T 0 )L 3,Pr = apple apple r u =0 @u @t +2E 1 ẑ u = rp + r 2 u + RaP r 1 T ẑ @T u z = Pr 1 r 2 T @t 23

3.3.2 Convection in a rotating plane layer T = T 0 L ~g T = T 1 Solutions to this set of equations are found by assuming the following form for the perturbations: {u,p,t} = {ũ(z), p(z), T (z)} exp( t +i(k x x + k y y)) Imposing of the (velocity) boundary conditions at the top and bottom wall will lead to a constraint: Instability will occur if = f(k x,k y,e,pr,ra)? = max k x,k y Re[ ] > 0 24

3.3.2 Convection in a rotating plane layer T = T 0 L ~g T = T 1 After a lengthy calculation, it is found for free-slip BCs that instability occurs if, Ra & 8.3E 4/3 and the wavelength k at the onset of instability scales as: k E 1/3 What s the temperature drop needed to have convection for Earth-like conditions? 25

3.3.2 Convection in a rotating plane layer Ra =8 10 7,E = 10 5,Pr =1 26

3.3.2 Convection in a rotating plane layer Ra =8 10 7,E = 10 5,Pr =1 Ra =1.5 10 9,E = 10 5,Pr =1 27

3.3.3 Rotating convection in spherical shells ion solution. '1 litative sketch of the marginally unstable convective motions in an int (Busse, 1970) (Busse and Carrigan, 1976) (Olson, 2007) In a rotating spherical shell heat is again transported by columnar convection. Consists of a series of slowly drifting columns. Well understood with combination of theory, experiments & simulation. 28

3.3.3 Rotating convection in spherical shells (N.B. Geometry in this example movie is not that in core, except maybe inside the tangent cylinder. Here we have a simple plane layer with cold top and hot bottom) Fundamental mechanism consists of pairs of cyclonic and anti-cyclonic cells which involve upwelling and downwellings respectively at the outer boundary. These transport heat from the hot ICB to the colder CMB. Mechanism involves viscosity breaking the Taylor-Proudman constraint allowing heat to be transported: viscosity enters the vorticity balance due to the small azimuthal and radial length scale (tall, thin columns) and also in the viscous (Ekman) boundary layer. 29

3.3.3 turbation Rotating solution. convection in spherical shells '1 Spiralling cyclonic and anticyclonic columnar convection cells transport heat in rapidly rotating spherical systems (Busse, 1970) 30

3.3.3 Rotating convection in spherical shells The process of viscosity balancing the rotational constraint and permitting radial heat transport can be studied by considering the radial component of the vorticity equation: 2Ω r u z = ν r ( 2 u) 2Ω D ν ( m D ) 3 => m ( ν 2ΩD 2 ) 1/3 (a) (b) Streamlines of convective flow in rapidly rotating spherical shell, (a) looking from outside shell (b) in equatorial plane. (From Zhang and Gubbins, 2000) 31

Crucial simplification: gravity acts perpendicular to the rotation axis! 32 3.3.4 Rotating convection: Annulus model Simple quantitative model of convection outside tangent cylinder in a spherical shell with constant slope of upper and lower boundaries (after Busse, 1970, 2002)

3.3.4 Rotating convection: Annulus model Take z component of the curl of the Navier Stokes equation, defining the z component of vorticity e ξ z = ẑ ( u) using cartesian co-ords and neglecting nonlinear advection effects: ξ z t 2Ω u z z = gα T x + ν 2 ξ z Thus with gravity in the y direction the buoyancy force can directly drive changes in the z component of vorticity. Integrating in the z direction and averaging over columns of height H: t 1 H H/2 H/2 ξ z dz 2Ω H [u z z=h/2 u z z= H/2 ]= gα H x H/2 H/2 Tdz+ ν H 2 H/2 H/2 ξ z dz Assuming the angle between the top and boundary slopes and the equatorial plane is small and defining as the tangent of that angle then by incompressibility at the boundaries we have the conditions: u z = η u y 33

3.3.4 Rotating convection: Annulus model ξ z Relabelling and T so these now represent z-averaged quantities, the z averaged axial vorticity equation now takes the form: ξ z t 4Ωη H u y = gα T x + ν 2 ξ z Since z dependence has been averaged out, and z component of velocity is assumed to be small, we can expand the velocity as: u = ψẑ so that ξ z = 2 H ψ and u y = ψ/ x. The z-averaged axial vorticity eqn along with the z-averaged heat eqn may therefore be written in non-dimensional form as: ξ z t + β ψ x = Ra T x + 2 ξ z Pr T t = ψ x + 2 T where Ra = gα Td3 κν, β = 4Ωη d 3 νh and Pr = ν κ 34

3.3.4 Rotating convection: Annulus model We look for solutions of these equations of the following form: ψ = A(sin πy)e i(kx ωt), T = B(sin πy)e i(kx ωt) where A and B are complex constants. These forms implicitly satisfy stress free and isothermal BC in the y direction. Note also that z = r 2 H = Substituting the trial solutions into the heat equation we find that: iω P r T =+ik ψ (π 2 + k 2 ) T or ik T = (π 2 + k 2 ) iωp r ψ Substituting this and the trial solns into the vorticity equation gives: _ iω(π 2 + k 2 )ψ + ikβ ψ = Ra ik @ 2 @x 2 + @2 @y 2 =( 2 + k 2 ) ik (π 2 + k 2 ) iωp r ψ + (π2 + k 2 ) 2 ψ Rearranging provides the complex dispersion relation: iω(π 2 +k 2 )(π 2 +k 2 iωp r)+ikβ (π 2 +k 2 iωp r) =k 2 Ra (π 2 +k 2 ) 2 (π 2 +k 2 iωp r). 35

3.3.4 Rotating convection: Annulus model Collecting the imaginary terms yields iω(π 2 + k 2 ) 2 + ikβ (π 2 + k 2 )=+(π 2 + k 2 ) 2 iωp r ω = or β k (π 2 + k 2 )(1 + Pr). Note that when β is +ive, the phase speed of the disturbance ω/k is positive (i.e. disturbance propagates prograde or eastwards for en Earth). β When thermal and viscous terms are neglected this simplifies to demonstrating that we simply have QG Rossby waves. ω = β k (π 2 + k 2 ) Collecting the real terms on the other hand yields ω 2 (π 2 + k 2 )Pr+ kβ ωpr = k 2 Ra (π 2 + k 2 ) 3 Substituting the expression for the frequency obtained above: Ra = (π2 + k 2 ) 3 k 2 + (β ) 2 Pr (π 2 + k 2 ) (1 + Pr) ( 1 1 (1 + Pr) ) 36

3.3.4 Rotating convection: Annulus model Noting that 1 1/(1 + Pr)=Pr/(1 + Pr) Ra = (π2 + k 2 ) 3 k 2 + this may be re-written as (β ) 2 Pr 2 (π 2 + k 2 )(1 + Pr) 2 Assuming that k 2 >> π 2. (necessary in order for viscosity to be able to balance Coriolis force in the vorticity equation) this simplifies to Ra = k 4 + (β ) 2 Pr 2 k 2 (1 + Pr) 2 Next to find the condition for the onset of convection, we differentiate this with respect to k, 4k 3 + 2k 3 (β ) 2 Pr 2 (1 + Pr) 2 = 0 This gives the condition for convection to begin: k = (β ) 1/3 Pr 1/3 2 1/6 (1 + Pr) 1/3. critical wavenumber for onset of convection 37

3.3.4 Rotating convection: Annulus model Under the same approximation frequency becomes ω = k 2 >> π 2. β k(1 + Pr) Substituting for the critical wavenumber gives ω = 2 1/6 Pr 1/3 (1 + Pr) 2/3 (β ) 2/3 the expression for the Finally, substituting for the critical k into the expression for Ra we find: Ra = Pr 4/3 2 4/6 (1 + Pr) 4/3 (β ) 4/3 + Pr2 2 1/3 (1 + Pr) 2/3 Pr 2/3 (1 + Pr) 2 (β ) 4/3 In summary, this analysis has demonstrated that : Ra c = C 1 (β ) 4/3, k c = C 2 (β ) 1/3, ω c = C 3 (β ) 2/3 where β = ( ) Ω 4η d 3 ν H i.e. as rotation rate or the viscosity the Ra needed for convection, the lengthscale of the flow and the frequency (drift rate). 38

3.3.5 Non-linear convection We have now established that the critical Rayleigh number Ra c above which convection occurs scales as: Ra c E 4/3 Applied to Earth s core conditions, this corresponds to an ICB/CMB (super-adiabatic) temperature contrast of the order of microkelvin. It is customary to quantify the vigor of convection in terms of the Nusselt number, which is the ratio between the total heat flux at the boundary and the conductive heat flux, e.g. for a plane layer: Nu = * z=top c P apple T L + = L T @T @z We now would like to establish scaling laws of the form: Nu = f(ra, P r, E) 40

3.3.5 Non-linear convection More vigorous convection leads to enhanced mixing, and thus we expect the fluid to become more isothermal. Near the boundaries however, the velocity has to tend to zero, and therefore the (advective) transport of heat is inhibited. Within these boundary layers, heat transport is mainly conductive. In non-rotating convection, the boundary layer thickness is such that it remains marginally stable, i.e.: (Ra/Ra c ) 1/3 z } } T 41

3.3.5 Non-linear convection A simple boundary layer argument for the non-rotating case leads to Nu Ra Ra c 1/3 (Ra Ra c ) Alternative theories, experiments and simulations suggest however different scaling laws, e.g.. Nu (Ra/R c ) 2/7 Nu / Ra 0.309 (Niemela et al., Nature 2000) 42

3.3.5 Non-linear convection (King et al., JFM 2012) Experiments in rotating fluids show the following behavior: - Close to onset, the Nusselt number scales as Nu Ra 3 E 4. This implies that the Ekman boundary layer is marginally stable. - For highly supercritical convection, the non-rotating scaling law is recovered:. 43

3.3.6 Thermal winds In rotating fluids, if large density variations are present (for example due to strong temperature anomalies) then flows or winds involving shear can result: u z = gα ( T r) 2Ω Considering the component of this in the east-west (azimuthal) direction using spherical polar co-ordinates, we can see how a large temperature gradient in the latitudinal direction can produce a shear flow in the axial direction. u φ z = gα 2Ω 1 r T θ Such thermal winds may exist in Earth s core, for example inside the tangent cylinder above and below the inner core. 44

3.4.1 Boundary slope and the tangent cylinder Inside Tangent Cylinder Inside TC, columnar fluid elements stretched as move away the rotation axis. (From Hide, 1966) Inside Tangent Cylinder Outside TC, columnar fluid elements compressed as move away from rotation axis. Tangent cylinder = imaginary cylinder just fitting round inner core. Due to Taylor-Proudman theorem, this acts a boundary that it is difficult for fluid parcels to pass through. 46

3.4.2 Stewartson (shear) boundary layers In reality, due to finite viscosity, the tangent cylinder (TC) barrier is not infinitely sharp. It consists of a viscous (shear) boundary layer where the fluid adjusts between the flow regime insider the TC and that outside TC. Known as the Stewartson boundary layer. The Stewartson boundary layer interacts with the ICB Ekman boundary layer in the equatorial plane. Each type of boundary layer has its own characteristic thickness. E 1/2 E 1/5 E 2/7 E 1/3 E 1/4 E 2/5 (From Stewartson 1966) Instabilities of the Stewartson and Ekman boundary layer likely important! Shear layer near K (Stewartson boundary Layer) 47

3.5 Summary: self-assessment questions (1) What are the equations governing core dynamics and can you describe what each term in these equation means? (2) What is geostrophy and can you derive the Taylor-Proudman theorem? (3) How do quasi-geostrophic (Rossby) waves operate? (4) What form does convection take in a rapidly-rotating spherical shell? (5) What form of boundary layers are present in the core? Next time: Influence of magnetic fields on core motions (Magnetic winds, Taylor s constraint, Alfven waves) 49

References - Aurnou, J., (2007) Planetary core dynamics and convective heat transfer scaling. Geophys. Astrophys, Fluid. Dyn., Vol 101, 327-345. - Busse, F.H., (2002) Convective flows in rapidly rotating spheres and their dynamo action. Phys. Fluids, Vol 14, 1301-1314. - Chandrasekhar, S. (1961) Hydrodynamic and Hydromagnetic Stability. Oxford University Press. - Gubbins, D., and Roberts, P.H., (1987) Magnetohydrodynamics of Earth s core, in Geomagnetism Volume 2, Ed. Jacobs, J.A., Chapter 1, pp1-182. - Jones, C.A., (2007) Thermal and Compositional convection in the outer core. In Treatise on Geophysics, Vol 8 Core Dynamics, Ed. P. Olson, Chapter 8.04, pp. 131-185. 50