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conductive heat transport

9.4. IS THERE CONVECTION IN THE EARTH'S MANTLE? 99

This estimate gives the ratio of the two modes of heat transport, but it does not help

us too much yet because we do not know the order of magnitude V of the ow velocity.

This quantity can be obtained from the Navier-Stokes equation of section 8.6:

@( v) + ( vv) = 2v + F again

(8:51)

@t r r

The force F in the right hand side is the buoyancy force that is associated with the ow

while the term 2 v accounts for the viscosity of the ow with viscosity coe cient . The

r

mantle of Earth's is extremely viscous and mantle convection (if it exists at all) is a very

slow process. We will therefore assume that the inertia term @( v)=@t and the advection

term ( vv) are small compared to the viscous term 2 v. (This assumption would

r r

have to be supported by a proper scale analysis.) Under this assumption, the mantle ow

is predominantly governed by a balance between the viscous force and the buoyancy force:

2 v = ;F : (9.30)

r

The next step is to relate the buoyancy force in the temperature perturbation T. A

temperature perturbation T from a reference temperature T0 leads to a density perturba-

tion from the reference temperature 0 given by:

T:

= (9.31)

;

In this expression is the thermal expansion coe cient that accounts for the expansion

or contraction of material due to temperature changes.

Problem b: Explain why for most materials > 0. A notable exception is water at

temperatures below 4 C.

Problem c: Write (T0 + T) = 0 + and use the Taylor expansion (2.11) of section 2.1

truncated after the rst order term to show that the expansion coe cient is given

=@T.

by = ;@

Problem d: The buoyancy forces is given by Archimedes' law which states that this force

equals the weight of the displaced uid. Use this result, (9.30) and (9.31) in a scale

analysis to show that the velocity is of the following order:

g TL2

V (9.32)

where g is the acceleration of gravity.

Problem e: Use this to derive that the ratio of the convective heat transport to the

conductive heat transport is given by:

g TL2

convective heat transport (9.33)

conductive heat transport

CHAPTER 9. SCALE ANALYSIS

100

The right hand side of this expression is dimensionless, this term is called the Rayleigh

number which is denoted by Ra:

g TL2 :

Ra (9.34)

The Rayleigh number is an indicator for the mode of heat transport. When Ra 1 heat

is predominantly transported by convection. When the thermal expansion coe cient is

large and when the viscosity and the heat conduction coe cient are small the Rayleigh

number is large and heat is transported by convection.

Problem f: Explain physically why a large value of and small values of and lead

to convective heat transport rather than conductive heat transport.

Dimensionless numbers play a crucial role in uid mechanics. A discussion of the Rayleigh

number and other dimesion-less diagnostics such as the Prandtl number and the Grashof

number can be found in section 14.2 of Tritton 60]. The implications on the di erent values

of the Rayleigh number on the character of convection in the Earth's mantle is discussed

in refs. 43] and 62]. Of course, if one want to use a scale analysis one must know the

values of the physical properties involved. For the Earth's mantle, the thermal expansion

coe cient is not very well known because of the complications involved in laboratory

measurements of the thermal expansion under the extremely high ambient pressure of

Earth's mantle 16].

9.5 Making an equation dimensionless

Usually the terms in the equations that one wants to analyze have a physical dimension

such as temperature, velocity, etc. It can sometimes be useful to re-scale all the variables

in the equation in such a way that the rescaled variables are dimensionless. This is

convenient when setting up numerical solutions of the equations, but in general it also

introduces dimensionless numbers that govern the physics of the problem in a natural

way. As an example we will apply this technique here to the heat equation (9.28).

Any variable can be made dimensionless by dividing out a constant that has the di-

mension of the variable. As an example, let the characteristic temperature variation be

denoted by T0 , the dimensional temperature perturbation can then be written as:

T = T0 T 0 : (9.35)

The quantity T 0 is dimesion-less. In this section, dimensionless variables are denoted with

a prime. Of course we may not know all the suitable scale factors a-priori. For example,

let the characteristic time used for scale the time-variable be denoted by :

t = t0 : (9.36)

We can still leave open and later choose a value that simpli es the equations as much

as possible. Of course when we want to express the heat equation (9.28) in the new time

variable we need to specify how the dimensional time derivative @=@t is related to the

dimensionless time derivative @=@t0 .

9.5. MAKING AN EQUATION DIMENSIONLESS 101

Problem a: Use the chain-rule for di erentiation to show that

@ =1 @ : (9.37)

@t @t0

Problem b: Let the velocity be scaled with the characteristic velocity (9.32):

2

v = g T0 L v0 (9.38)

and let0 the position vector be scaled with the characteristic length L of the system:

r = Lr . Use a result similar to (9.37) to convert the spatial derivatives to the new

space coordinate and re-scale all terms in the heat equation (9.28) to derive the

following dimensionless form of this equation

L2 @T 0 + g T0 L3 v0 T 0 = 2T 0

0 0

(9.39)

@t0 r r

is the gradient operator with respect to the dimensionless coordinates r0 .

where 0

r

At this point we have not speci ed the time-scale for the scaling of the time variable

yet. The equation (9.39) simpli es as much as possible when we choose in such a way

that the constant that multiplies @T 0 =@t0 is equal to unity:

= L2 = : (9.40)

Problem c: Suppose heat would only be transported by conduction: @T=@t = 2 T.

r

Use a scale analysis to show that given by (9.40) is the characteristic time-scale

for heat conduction.

This means that the scaling of the time variable expresses the time in units of the

characteristic di usion time for heat.

Problem d: Show that with this choice of the dimensionless heat equation is given by:

@T 0 + Rar0 v0 T 0 = 2T 0

0

(9.41)

@t0 r

where Ra is the Rayleigh number.

The advantage of this dimensionless equation over the original heat equation is that

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