Rotating fluids and the outer core - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 404

The viscous stress arising through the bulk viscosity , ζ, depends on the compres-
sion or dilatation rate. For its determination, one requires either a very compress-
ible fluid or very high flow pressures. Few satisfactory measurements of ζ have
been made for any fluid. For the Earth's core, the damping of compressional waves
indicates that ζ is quite small. The dynamic viscosity , η, has traditionally been
estimated on the basis of laboratory studies to be in the order of 10 − 2 Pa s (Poirier,
1988). Such extrapolation to core pressures and temperatures has been challenged
by Brazhkin (1998) and Brazhkin and Lyapin (2000), who find 10 11 Pa s at the bot-
tom of the outer core and 10 2 Pa s at the top. These values are closely confirmed
by the reduction in the rotational splitting of the equatorial translational modes
of the inner core (Smylie, 1999), giving 1.22 10 11 Pa s at the bottom, and by

the free decay of the free core nutations (Palmer and Smylie, 2005; Smylie and
Palmer, 2007), leading to 890-3900 Pa s at the top of the outer core. Direct obser-
vations have recently been reconciled with laboratory measurements by Smylie
et al. (2009), taking into account the pressure dependence of the activation volume,
which had been ignored in previous extrapolations of laboratory measurements.
In addition to the contribution of viscous terms to the stress field, we must add
pressure forces, which contribute the diagonal terms

×

i

j to the stress tensor.
Substituting these and expression (6.115) for the viscous stress in the equation
of motion (6.103) yields

− p δ

∂v j
∂ x i +

∂v i

∂ t + v j ∂v i

1
ρ

∂
∂ x j

∂v i
∂ x j −

2
3 ∇·

i
j

i i

j

∂ x j =

f i + − p δ + η v δ + ζ ∇· v δ

j

2

1
ρ ∂ x i + η

∂ p

∂ x j ∂ x j + ζ + η/ 3

∂ v i ∂

∂ x i

= f i − ( ∇· v ), (6.116)

ρ ρ

ignoring spatial gradients of the coe cients of viscosity. Also left unspecified is

the body force per unit mass, f i . In the uniformly rotating reference frame, it will
include the negatives of the Coriolis acceleration (3.75) and the centrifugal accel-
eration (3.76). As well, in the self-gravitating fluid outer core, it will include the
gravitational acceleration g . In symbolic notation, the equation of motion becomes

∂ v
∂ t +

( v ·∇ ) v +

2 Ω × v

ζ + η/ 3
ρ

1
ρ ∇

η
ρ ∇

2 v

=− p + g − Ω × ( Ω × r ) + + ∇ ( ∇· v ). (6.117)

We next examine the energy equation. Again, we enclose the volume V by the

surface S moving with the fluid. The change in the internal energy per unit mass is

Tds − pd v, where T is the temperature, s is the entropy per unit mass or the specific