Theoretical Geomagnetism

Lecture 3

Core Dynamics I: Rotating Convection in Spherical Geometry

1

• Thermal (and chemical buoyancy) produces convection.

3.0 Ingredients of core dynamics

• Rotation places constraints on motions.

• Dynamo generated magnetic field will also influence motions.

• Spherical shell geometry.

2

3.0 Ingredients of core dynamics

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of buoyancy-driven flows and how such convective motions can generate dynamoaction. In addition to buoyancy forces, core flows are also strongly affected byCoriolis forces and Lorentz forces. In cases where the Coriolis forces dominate overthe Lorentz forces, fluid motions tend to become two-dimensional along the directionof the rotation axis (see figure 1). It is estimated that Coriolis and Lorentz forcestend to be comparable in planetary core settings based on theoretical arguments(e.g. Fearn 1998) and from extrapolations of planetary magnetic field observations(e.g. Stevenson 2003). However, numerical simulations have shown that nearlytwo-dimensional, axially-aligned motions persist even for strong magnetic fields, withLorentz forces up to !10 times stronger than Coriolis forces (Olson and Glatzmaier1995, Busse 2002). Thus, it is reasonable to assume that non-magnetic rotatingconvection experiments provide insight into the dynamics of planetary core processes.

The non-dimensional parameters that describe thermal convection in rapidly rotatingspherical shells are defined in table 1. Typical parameter values in present-day modelsare far from the estimated values for planetary cores. However, it maybe be possibleto simulate the convection dynamics without having to replicate the planetary coreparameter values. Many physical systems have well-defined, limiting ‘asymptotic’behaviors at either large or small values of the control parameters. If models showthat a system has such a limiting scaling behavior, then extrapolation to planetaryparameter values may be justified.

For example, the schematic in figure 2 shows a typical function f that varies withcontrol parameter x. For intermediate values of x, the function f(x) varies stronglywith x. Hexagonal symbols mark out a set of hypothetical experimental results for

Figure 1. Schematic showing the two basic convective flow structures thought to exist in planetary cores:large-scale axisymmetric zonal flows and small-scale convection columns.

328 J. M. Aurnou

Small Scale Convection Columns

Magnetic FieldTangent cylinder

barrier to flow

(Modified from Aurnou, 2007) 3

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

Lecture 3: Core Dynamics I

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

5

(known together as flow inertia i.e. the acceleration of a fluid parcel)

Du

Dt

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.

• is now the fluid velocity in the rotating reference frame.u

6

3.1.3 The 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 with a background temperature field T0, and perturbations T evolving as:

- 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 as described in the buoyancy force.

∂T

∂t+ (u ·∇)T0 = κ∇2T

ρ0

g

ραT

ρ∂u

∂t+ ρ(u ·∇)u + 2ρ(Ω × u) = −∇P + ραTg + ρν∇2

u00 00 0

Temperature differences produce a change in density and so a buoyancy force.

_

7

3.1.4 The Lorentz force

∂B

∂t= ∇× (u × B) + η∇2

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,

ρ∂u

∂t+ ρ(u ·∇)u + 2ρ(Ω × u) = −∇P + ραTg + ρ(J × B) + ρν∇2

u0 0 00 0

Lorentz force: Force due to magnetic field acting on electric currents in fluid.

_

JB =1

µ0(rB)B = r

B2

2µ0

+

1

µ0(B ·r)B

8

3.1.5 Equations governing core dynamics

• To get some feeling for the flow patterns and dynamics possible for such a system, we will consider some simple examples.....

• These are the equations of convection-driven, rotating magnetohydrodynamics under the Boussinesq approx.

+ Boundary Flow, Thermal and Magnetic BC on ICB & CMB.

∂T

∂t+ (u ·∇)T0 = κ∇2T

∂B

∂t= ∇× (u × B) + η∇2

B

∇ · u = 0

∇ · B = 0

ρ∂u

∂t+ ρ(u ·∇)u + 2ρ(Ω × u) = −∇P + ραTg + ρ(J × B) + ρν∇2

u0 0 00 0

Terms representing feedback and coupling between the velocity, magnetic and temperature fields

_

9

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

Lecture 3: Core Dynamics I

10

• 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:

3.2.1 Geostrophic Balance

2(Ω × u) = −∇P

ρ Geostrophic Balance between the Coriolis force and the pressure gradient.

• Flow is perpendicular to pressure gradient and involves circulations around regions of high and low pressure.

L

u

• Ω

∇P

11

3.2.2 Taylor-Proudman Theorem• Assuming the rotation vector is parallel to the z axis so ,Ω = Ωz

2(Ωz × u) = −∇P

ρ

• Taking the curl of both sides,

2Ω∂u

∂z= −

1

ρ2(∇P ×∇ρ)

• If density gradients are negligible so that ,ρ = ρ0

∂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

10

Columnar Motions

• Spun-up cylindrical tank of fluid in solid-

body rotation

• Ekman ~ 3 x 10-5

» 10 cm thick water layer & 33 rpm

• For slow (low Ro), quasi-steady(low F), rapidly

rotating (low E) flows, we expect TPT to hold

– Fluid should move around topography in order

to remain on geostrophic contours

How to Form a Taylor Column?

Non-rotating case

!

"u

"z# 0

Obstacle

11

How to Form a Taylor Column?

Rapidly-rotating flow past obstacleRapidly-rotating case

!

"u

"z= 0!

Obstacle

Taylor Column

Taylor-Proudman

Theorem

Taylor Column: The Movie

Movie

: C

. S

afr

anek-S

chra

der

(From Aurnou, Les Houches Lectures, 2007)

(From Brenner, 1999)

GI Taylor

13

• For incompressible fluids, in order for then

3.2.3 Geostrophic motions∂u/∂z = 0

• This requires z-independent flow patterns motions to follow contours of constant depth.

e.g. Flow patterns move only zonally at fixed latitude in a spherical shell without changing height H.

H

Geostrophic Contours

Ω

∂ux

∂x+

∂uy

∂y= 0

14

• Forcing by convection, inhomogeneous boundary conditions or magnetic effects will lead to departures from perfect geostrophy.

3.2.4 Quasi-geostrophic motions

• 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

∂t+ (u ·∇)u + 2(Ω × u) = −

∇P

ρ0

• This can be re-written in the form of a ‘frozen flux’ eqn for ,

(1)D(ξ + 2Ω)

Dt= [(ξ + 2Ω) ·∇]u

ξ + 2Ω

• Curling this we obtain an evolution eqn for the vorticity :ξ = ∇× u

∂ξ

∂t+ (u ·∇)ξ − (ξ ·∇)u − 2Ω

∂u

∂z= 0

15

• 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 direction:

u

z

u = u(x, y)x + v(x, y)y + w(x, y, z)z where w << u, v

• Substituting into (1), then to leading order the (axial) component of the vorticity equation is,

D(ξz + 2Ω)

Dt= (ξz + 2Ω)

∂w

∂z

z

3.2.4 Quasi-geostrophic motions

• The approximately 2D flow can therefore be represented in terms of a stream-function representation in terms of a single scalar:

Rac = C1(β∗)4/3, kc = C2(β

∗)1/3, ωc = C3(β∗)2/3 where β∗ =

(Ω

ν

)4η∗d3

H(27)

∂u

∂z= −

gα

2Ω(∇× T r) (28)

∂uφ

∂z=

gα

2Ω

1

r

∂T

∂θ(29)

(u, v, 0) =

(∂ψ

∂y,−

∂ψ

∂x, 0

)

(30)

4

16

10– potential vorticity conservation

• We now integrate the scalar vorticity equation over the height H of a fluid column, in aclosed container. We temporary omit the energy source (forcing) and sink (viscous)terms:

Hddt

( + 2) = ( + 2)(w(H) w(0))

w(H)

w(0)

uH

Due to non-penetration at the boundaries, w(H) w(0) isthe variation in height of the column as it evolves:

w(H) w(0) =dHdt

the vorticity equation then writes

Hddt

( + 2) ( + 2)dHdt= 0

which yieldsddt

+ 2

H

= 0

• In an inviscid rotating fluid without forcing, the potential vorticity q = + 2

His materially

conserved.

• a compressed column acquires anticyclonic vorticity (Jon’s droplet experiment). Astretched column acquires cyclonic vorticity.

(From J. Aubert, Les Houches lecture, 2007)

• Due to non-penetration at the boundaries,

• i.e. in the absence of forcing or dissipation, potential vorticity is conserved moving with the fluid.

• Integrating over the direction, this yields an expression depending on the axial flow at the boundaries,

z

HD(ξz + 2Ω)

Dt= (ξz + 2Ω)[w(H) − w(0)]

• So, H

D(ξz + 2Ω)

Dt− (ξz + 2Ω)

DH

Dt= 0

• Or,D

Dt

(ξz + 2Ω

H

)

= 0 (2)

3.2.4 Quasi-geostrophic motions

DH

Dt= [w(H) w(0)]

17

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

Lecture 3: Core Dynamics I

18

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

20

L ~g

T = T0

T = T1

• The equations describing the physical behavior of this system are:

• A purely conductive solution is:

~

r · u = 0

@u

@t+ u ·ru+ 2z u

= rp+ r2u+ ↵Tgz

@T

@t+ u ·rT = r2T

u = 0, T = T1 + (T0 T1)z

L, p = ↵g

T1z + (T0 T1)

z2

2L

3.3.2 Convection in a rotating plane layer

21

L ~g

T = T0

T = T1

• 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 )

u = U + "u0, p = P + "p0, T = + "T 0

" 1

~

3.3.2 Convection in a rotating plane layer

22

L ~g

T = T0

T = T1

• Neglecting terms in w.r.t. and omitting the ‘, we obtain the following (linear) perturbation equations:

"2 "

r · u = 0

@u

@t+ 2z u

= rp+ r2u+ ↵gT z

@T

@t+ u ·r = r2T

~

3.3.2 Convection in a rotating plane layer

23

L ~g

T = T0

T = T1

• It’s customary to write these equations in non-dimensional form with

E =

L2, Ra =

↵g(T1 T0)L3

, P r =

r · u = 0

@u

@t+ 2E1z u = rp+r2u+RaPr1T z

@T

@t uz = Pr1r2T

~

3.3.2 Convection in a rotating plane layer

24

L ~g

T = T0

T = T1

• Solutions to this set of equations are found by assuming the following form for the perturbations:

• Imposing of the (velocity) boundary conditions at the top and bottom wall will lead to a constraint:

• Instability will occur if

u, p, T = ˜u(z), p(z), ˜T (z) exp(t+ i(k

x

x+ k

y

y))

= f(kx

, ky

, E, Pr,Ra)

?= max

kx

,ky

Re[] > 0

3.3.2 Convection in a rotating plane layer

25

L ~g

T = T0

T = T1

• After a lengthy calculation, it is found for free-slip BCs that instability occurs if,

and the wavelength k at the onset of instability scales as:

• What’s the temperature drop needed to have convection for Earth-like conditions?

Ra & 8.3E4/3

k E1/3

3.3.2 Convection in a rotating plane layer

26

Ra = 8 · 107, E = 105, P r = 1

3.3.2 Convection in a rotating plane layer

27

Ra = 8 · 107, E = 105, P r = 1 Ra = 1.5 · 109, E = 105, P r = 1

• In a rotating spherical shell heat is again transported by columnar convection.

3.3.3 Rotating convection in spherical shells

• Well understood with combination of theory, experiments & simulation.

Thermal ~ n ~ ~ a b i l i ~ i e s in rapidly rotating systems 453

number unless the analysis is restricted to the axisymmetric case. For this reason Roberts's analysis is repeated in 9 5 for modes which have the same symmetry as the perturbation solution.

'1 FIGURE 1. Qualitative sketch of the marginally unstable convective motions in an internally

heated rote ting sphere.

5. Comparison with the general asymptotic theory In the case of a finite change in depth, the velocity component parallel to the

axis of rotation is no longer negligible in comparison with the perpendicular components. In order to analyse the basic equations (4.1)-(4.3), we introduce, as a representation of the solenoidal vector field v,

v = V x ( V x k v ) + V x k w .

The elimination of the pressure from (4.1) yields the following equations for v, w and 9, equivalent to the system (4.1)-(4.3):

(5.1) 1 (EV2-iO)V2A2v-2k.VA,w-gk. ( rV29-V(r .V+ 1)8) = 0,

(EV2-iG)A2w+2k.VA,v+&kxr.V9 = 0,

(V2-PE-liO)9-r.Vx(kxV)v+kxr.Vw = 0.

As before, the operator A, is used as abbreviation for the operator V2 - (kV2). In the limit of high rotation rates, it can be assumed that the characteristic scale of the solution in the p- and $-directions of a cylindrical system of co-ordinates

theoretical, experimental, and numerical studies ofconvection in rapidly rotating spherical shells give usvaluable insight into how it works. Planetary rotationaffects core convection through the so-calledTaylor–Proudman constraint, where the local axisof fluid vorticity tends to align parallel (or anti-parallel) to Earth’s spin axis (Taylor, 1923). Theresulting convective motion, shown in Figure 13, isoften called ‘columnar’, because the variation in flowstructure parallel to the spin axis is smaller than thevariations in perpendicular planes, that is, parallel tothe equatorial plane. Theory indicates that theLorentz force due to the geomagnetic field withinthe core distorts the columns in the equatorial plane,as does the compressibility of the core fluid and thecurvature and roughness of the core–mantle andinner-core boundaries. Columnar convectionincludes secondary motions with components paral-lel to the spin axis. The combination of thesesecondary motions and the primary geostrophicmotion makes convection helical, in which the vor-ticity and velocity vectors are correlated on average,being generally negative in the NorthernHemisphere of the outer core and generally positivein the Southern Hemisphere. This helical motiongives rise to the aforementioned !-effect throughwhich poloidal magnetic field is induced from toroi-dal magnetic field, and vice versa. The forward andbackward transformation of magnetic energy

between these two types of fields in the outer coreis the key kinematic ingredient in the geodynamo.Helicity is not just a property of convection; otherrotationally dominated flows, such as the flowsinduced by precession, are helical. Another inductionmechanism is provided by large-scale toroidal flows,such as thermal winds and the flows associated withdifferential rotation of the inner core. The shear inthese nearly horizontal flows converts poloidal mag-netic field lines into toroidal magnetic field lines,much like windings on a ball of string, providingthe so-called ‘! -effect’. The relative contributionsof the ! and ! effects in the geodynamo are uncer-tain, because the strength of the toroidal field in thecore is unknown. If the toroidal field is only about asstrong as the poloidal field in the core (1–10mT),then the geodynamo is likely an !2-type dynamo,and the ! effect due to large-scale shear flows doesnot control the field strength in the core. Muchstronger field strengths in the core would imply alarger role for the ! effect, and the geodynamo wouldthen be classified as !! or !2!. It is significant thatboth effects are functions of latitude, offering a fun-damental explanation for why the geomagnetic fieldtends toward an axially symmetric dipole time-aver-age configuration.

An important dynamical issue is the physics ofmagnetic field equilibration – what controls the geo-magnetic field strength. In this context, theoristsdistinguish between two generic classes of dynamos,called ‘weak’ and ‘strong’, respectively. In strong fielddynamos, the Lorentz force exerted by the magneticfield on the fluid is comparable to the Coriolis accel-eration, whereas in weak-field dynamos the Lorentzforce is secondary. The fluid motion in strong fielddynamos is sometimes called ‘magnetostrophic’, indi-cating the tendency for the fluid to circulate around,rather than directly crossing, the magnetic field lines.In strong field dynamos, the Elsasser number is oforder one. Assuming the geodynamo is a strong fielddynamo, an Elsasser number of one implies an rootmean square (rms) field of about 4mT in the core.Strong-field dynamos tend to equilibrate by tradingoff magnetic and kinetic energy. Increases in mag-netic energy increase the Lorentz force, reducing thekinetic energy of the fluid, which ultimately causesthe field to decrease back toward its equilibriumstrength. Conversely, decreases in magnetic energydecrease the Lorentz force, allowing the kineticenergy to increase, which induces a stronger mag-netic field. This situation would seem to provide astable equilibrium field strength (given a constant

Figure 13 Columnar-style convection in a rotatingspherical shell. Numerical model results show columnarconcentrations of axial vorticity (red¼positive,blue¼ negative vorticity).

Overview 15

(Busse, 1970) (Olson, 2007)(Busse and Carrigan, 1976)

• Consists of a series of slowly drifting columns.

28

• 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.

(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)

3.3.3 Rotating convection in spherical shells

29

Thermal ~ n ~ ~ a b i l i ~ i e s in rapidly rotating systems 453

number unless the analysis is restricted to the axisymmetric case. For this reason Roberts's analysis is repeated in 9 5 for modes which have the same symmetry as the perturbation solution.

'1 FIGURE 1. Qualitative sketch of the marginally unstable convective motions in an internally

heated rote ting sphere.

5. Comparison with the general asymptotic theory In the case of a finite change in depth, the velocity component parallel to the

axis of rotation is no longer negligible in comparison with the perpendicular components. In order to analyse the basic equations (4.1)-(4.3), we introduce, as a representation of the solenoidal vector field v,

v = V x ( V x k v ) + V x k w .

The elimination of the pressure from (4.1) yields the following equations for v, w and 9, equivalent to the system (4.1)-(4.3):

(5.1) 1 (EV2-iO)V2A2v-2k.VA,w-gk. ( rV29-V(r .V+ 1)8) = 0,

(EV2-iG)A2w+2k.VA,v+&kxr.V9 = 0,

(V2-PE-liO)9-r.Vx(kxV)v+kxr.Vw = 0.

As before, the operator A, is used as abbreviation for the operator V2 - (kV2). In the limit of high rotation rates, it can be assumed that the characteristic scale of the solution in the p- and $-directions of a cylindrical system of co-ordinates

Spiralling cyclonic and anti-cyclonic columnar convection cells transport heat in rapidly rotating spherical systems (Busse, 1970)

3.3.3 Rotating convection in spherical shells

30

• We have now established that the critical Rayleigh number above which convection occurs scales as:

3.3.5 Non-linear convection

• Applied to Earth’s core conditions, this corresponds to an ICB/CMB (super-adiabatic) temperature contrast of the order of microkelvin.

40

Rac

Rac E4/3

• 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:

• We now would like to establish scaling laws of the form:

Nu =

*|

z=top

cP

T

L

+= L

T

@T

@z

Nu = f(Ra, Pr,E)

• In non-rotating convection, the boundary layer thickness is such that it remains marginally stable, i.e.:

• More vigorous convection leads to enhanced mixing, and thus we expect the fluid to become more isothermal.

3.3.5 Non-linear convection

• 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.

41

(Ra/Rac)1/3

T

z

J.Noir (ETHZ) - Introduction to GAFD2

What is the thickness of the boundary layer ?

• Express the thickness d as a function of the critical Rayleigh number Rac.

• Assuming that the same heat flux goes thru the top and bottom boundary layer and that the temperature is uniform in the interior, express the temperature gradient across the thermal boundary layers DT as a function of T1 and T2.

T1

T1

• Demonstrate that

J.Noir (ETHZ) - Introduction to GAFD3

T1

T1

What is the thickness of the boundary layer ?

J.Noir (ETHZ) - Introduction to GAFD4

What is the heat flux through the boundary layers ?

You know that all the heat coming from the bottom must travel vertically and leave throughs the top surface.

T1

T1

• Demonstrate that

Express the Nusselt number as a fonction of d, DT, H and the thermal conductivity k of the fluid

Nu /

Ra

Rac

1/3

J.Noir (ETHZ) - Introduction to GAFD5

One particular case where Nu can be estimated

Qtotal = QBL

We know that all the heat that has been advected upward must traverse the boundary layers at the top and bottom:

Q

J.Noir (ETHZ) - Introduction to GAFD6

One particular case where Nu can be estimated

Q

Nu /

Ra

Rac

1/3

QBL kT/2

= k

T/2

H

Ra

Rac

1/3

Qconduction kT

H

3.3.5 Non-linear convection

43

• Experiments in rotating fluids show the following behavior:- Close to onset, the Nusselt number scales as . This implies that the Ekman boundary layer is marginally stable.- For highly supercritical convection, the non-rotating scaling law is recovered: .

Nu Ra3E4

(King et al., JFM 2012)

• In rotating fluids, if large density variations are present (for example due to strong temperature anomalies) then flows or winds involving shear can result:

3.3.6 Thermal winds

• 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.

Rac = C1(β∗)4/3, kc = C2(β

∗)1/3, ωc = C3(β∗)2/3 where β∗ =

(Ω

ν

)4η∗d3

H(27)

∂u

∂z= −

gα

2Ω(∇× T r) (28)

∂uφ

∂z=

gα

2Ω

1

r

∂T

∂θ(29)

4

• Such ‘thermal winds’ may exist in Earth’s core, for example inside the tangent cylinder above and below the inner core.

44

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

Lecture 3: Core Dynamics I

45

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

3.6 Summary

Lecture 3: Core Dynamics I

48

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

- 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.

- Gubbins, D., and Roberts, P.H., (1987) Magnetohydrodynamics of Earth’s core, in Geomagnetism Volume 2, Ed. Jacobs, J.A., Chapter 1, pp1-182.

- 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.

- Aurnou, J., (2007) Planetary core dynamics and convective heat transfer scaling. Geophys. Astrophys, Fluid. Dyn., Vol 101, 327-345.

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