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14. Convection Theory; Adiabats and Boundary Layers
Thermal Convective Instability
We saw in the last chapter that in the presence of an unstable vertical density
profile and no diffusive decay of the entity (heat, composition) responsible
for it, there is a “direct” instability, i.e. all disturbances grow exponentially
in time. When limited by viscosity, the characteristic timescale for
exponentiation is about ν /g(δρ / ρ)L , where the density anomaly is density
gradient times characteristic lengthscale (i.e. δρ / ρ ~ β /k; L ~ k −1). The exact
definition of an unstable density profile must of course be given in the
following way: Does an element of fluid, upon vertical displacement, find
itself in an environment such that its buoyancy causes it’s motion to be
accelerated? Clearly, this must ignore adiabatic expansions and contractions
that arise because of the displacement to a new pressure; buoyancy (positive
or negative) arises from deviations from adiabaticity or non-uniform
composition. In this context, “adiabatic” means isentropic.
In the specific case of thermal effects, we need to take into account the fact
that the thermal anomalies are not merely advected but also spread
diffusively. So let’s now do this. The heat diffusion equation including
advection and assuming incompressible flow can be written:

∂T
Q
= κ∇ 2T − u.∇T +
∂t
Cp
(14.1)
where Q is the heat generation per unit mass. Let’s look now at the behavior
of perturbations θ to the temperature field, assuming that the background
state takes the form of a uniform temperature gradient relative to the adiabat
; i.e. T = Tad -βz + θ. (Notice that β is now a temperature gradient whereas
previously we used it to express a density gradient. If positive, it represents
an unstable case, assuming of course that the coefficient of thermal
expansion α>0. Notice that β is not the actual temperature gradient because
Tad is a function of z. In some cases, β will be tiny compared to dTad/dz!)
The equation for linear perturbations becomes:
∂θ
= κ∇ 2θ + uzβ
∂t
(14.2)
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(where, as always, non-linear terms are neglected because all the
perturbation variables are infinitesimal). The associated density anomalies
are:
δρ = −ρ 0αθ
(14.3)
where α is the coefficient of thermal expansion. Treating temperature
anomalies, etc. as varying as exp(ik.r+σt) as before, we get that
(σ + κk 2 )θ = uzβ
(14.4)
Looking back now at our analysis of the vorticity equation 13.20, we have:
gαβ (k x + k y )
(σ + νk )(σ + κk ) =
k2
2
2
2
2
(14.5)
This has a different behavior from eqn 13.25 (RT instabilities): Positive
solutions for σ are no longer assured for all positive values of the other
parameters. Clearly, the onset of the instability (which you can think of as
the lowest value of β for which σ≥0) is found by setting σ=0, and this
corresponds to:
gαβ (k x + k y )
=1
νκk 6
2
2
(14.6)
But we can still ask about the best choice of wavevector. In a real system
you must satisfy boundary conditions, and for a fluid confined between z=0
and z=L (say), this enforces a choice of π/L for the vertical component of the
wavevector. (Actually, this is not obvious, but it turns out to be exactly so
for “free” boundaries, meaning those for which the component of flow
parallel to the boundaries is unimpeded). But we are at liberty to choose the
horizontal components (if our fluid is not in a box with sidewalls). So the
eigensolutions of interest have the form:
u(x,y,z) = A sin (kxx) cos (kyy) cos(πz/L)
v(x,y,z) = B cos (kxx) sin (kyy) cos(πz/L)
w((x,y,z) = C cos (kxx) cos (kyy) sin(πz/L)
(14.7)
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for x, y and z components of the velocity field. A, B and C are arbitrary but
small and must satisfy Akx+Bky+C(π/L) =0 to assure divergence-free flow.
Notice that the chosen form satisfies the boundary conditions.
The lowest value of temperature gradient will be achieved for the choice of
wavevector that maximizes the quantity y/(1+y)3 where y is defined as
2
2
2
(k x + k x ) /k z . This value of y is 1/2 (which means that the characteristic
wavelength of the horizontal variation in temperature, etc. is 23/2L) and the
resulting value of y/(1+y)3 is 4/27. The relative values of kx and ky are
arbitrary.
(1+y)3/y
≡gαβ(L/π)4/ νκ
27/4
Of course, this is the same as minimizing the value of (1+y)3/y, which is the
graph shown. So putting this all together, we get
27π 4
= 657.5..
4
gαβL4
Ra ≡
νκ
Rac =
(14.8)
where Ra is called the Rayleigh number and the subscript “c” refers to its
critical value (the value below which no motion will be amplified). The
“proof” above is actually technically incorrect, because we have not
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explicitly established that our solution satisfies the boundary conditions, but
it turns out to be correct nonetheless for free boundaries. The critical
Rayleigh number is different for other boundary conditions but typically of
order a thousand. The system we have analyzed is called Rayleigh-Benard
convection, because the “starting state” of a uniform temperature gradient is
appropriate to heating a fluid from below.
Notice that this result applies for both low viscosities and high viscosities.
But the quantitative difference is enormous for planetary scales (large L).
Physical Interpretation
As our discussion of the Rayleigh-Taylor problem revealed, there is a
characteristic timescale for growth of the flow. It is of order ν /g(δρ / ρ)L . But
of course, this must be shorter than the characteristic diffusion time L2 /κ
since otherwise the instability will be shut off. Clearly the ratio of timescales
gives something with the ingredients of the Rayleigh number. So the
Rayleigh number can be thought of as the ratio of timescales (thermal
timescale divided by dynamic timescale). If it is too small then the thermal
diffusion will win out over the buoyancy driven flow and the disturbance
will not grow (i.e., convection will not occur).
Application of the Rayleigh Number
Example # 1: Low Viscosity Suppose we have L~1000km, ν~κ~ 0.01cm2
/sec, α~ 10-5 K-1 and g~1000 cgs, then Ra ~ 1000 requires a superadiabatic
temperature gradient of about 10-31 K/cm, or a temperature excess across L
of 10-23 K! Actually this is misleading because convection in low viscosity
systems is usually strongly affected by rotation (and sometimes by magnetic
fields) but the point is nonetheless clear..... ridiculously small temperature
excesses are required to set up convection in low viscosity systems. This is
part of the arguments that leads eventually to the conclusion that if a low
viscosity, homogeneous fluid is convecting, it will be in an adiabatic state not because it is actually adiabatic, but because the deviations from
adiabaticity are unmeasurably small.
Example # 2: High Viscosity Now take the same numbers as above except
that the viscosity is 24 orders of magnitude larger. We then require a
temperature excess (on top of the adiabat, remember) of order 10K. This is
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beginning to be significant. So in sub-solidus convection, we do have to be
concerned about the temperature anomalies that drive convection, and they
have detectable consequences (geoid, seismic tomography).
Finite Amplitude Convection
It is important to understand that the above analysis tells you only if
convection occurs; it tells you nothing about the form that the convection
actually takes. The reason is that once the convective heat flux becomes
significant, the assumptions made in the above analysis are no longer valid.
There is no rigorous analytical theory for finite amplitude convection
(except near convective onset) because it is a non-linear phenomenon and
chaotic at sufficiently high (“interesting”) amplitudes. It is of course possible
to do numerical or laboratory experiments.
A. Inviscid Convection
In the limit where viscosity does not matter, the following simple picture
known as mixing length theory does a reasonable job characterizing the
convection. In this context, “reasonable” means order-of-magnitude. We
imagine that the hot rising fluid elements and cold sinking fluid elements are
“destroyed” by turbulence (what this really means is that there is a cascade
of the kinetic energy into smaller scale incoherent motions). We can treat
these elements as rising under the action of an effective gravity gαβL, where
L is the large length scale in the problem (the mixing length, and some large
fraction of the dimensions of the system). So the velocity attained by a fluid
blob going a distance L is accordingly (by Newton’s laws):
Vconv ≈ gαβL2
(14.9)
and the heat flow carried by these upwellings and downwellings is
Fconv ≈ ρC pVconv βL
(14.10)
If we recall from our thermodynamics that the characteristic temperature
scale height for an adiabatic fluid is HT ≡T/(-dT/dz)ad = Cp/αg, then it
follows that
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⎡ LF ⎤
≈ 0.1⎢ conv ⎥
⎣ ρHT ⎦
1/ 3
Vconv
2
10 ⎛ Vconv ⎞
βL ≈
⎜
⎟
α ⎝ gL ⎠
(14.11)
2
where the numerical coefficients are uncertain; some would say that the
correct value is more like unity. Here are some typical numbers for Earth’s
core and for Jupiter’s interior:
For Earth’s core, we have plausibly F~10 cgs, L~108 cm, L/H ~0.1, ρ~10
and we get V~0.05 cm/sec, not too different from estimates based on
variations in the magnetic field. The characteristic scale velocity (gL)1/2 ~ 3
x 105 cm/s which is almost seven orders of magnitude larger. Accordingly,
the excess temperature drop (βL) is very small, about (100).(2 x 10-14 )/(5 x
10-6 ) ~ microKelvin. [Actually, a more careful analysis involving rotation
and magnetic field suggests something closer to a milliKelvin, but it is small
anyway!]. In this case, the actual temperature is mostly adiabatic and the
temperature gradient is therefore many orders of magnitude larger than the
superadiabatic part that drives convection. For example, the actual
ltemparature difference across Earth’s outer core is roughly 1000K but the
part of this that drives convection is less than a millikelvin. This general
picture is confirmed in numerical experiments.
For Jupiter’s core, we have plausibly F~3000 cgs, L~109 cm, L/H ~0.1, ρ~1
cgs and we get V~1 cm/sec. The characteristic scale velocity (gL)1/2 ~ 2 x
106 cm/s which is over six orders of magnitude larger. Accordingly, the
excess temperature drop (βL) is very small, about (100).(2 x 10-13 ) / (5 x 106
) ~ ten microKelvin or so. This is qualitatively like Earth’s core.
If the low viscosity fluid is bounded by a “brick wall”(e.g. core-mantle
boundary in Earth’s case) then one must do a separate analysis to understand
that region. However, the thickness of this layer is very small when the
viscosity is low, so the conclusion about small temperature drops (relative to
an adiabat) still applies.
When does the assumption that “the viscosity doesn’t matter” break down?
Looking back at the equation of motion, we require that the ratio of inertial
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to viscous terms is small, or in other words V/(L/V) > νV/L2. The ratio of
these terms is VL/ν and is called the Reynolds number.
Re≡ VL/ν
(14.12)
For the values quoted above, we would need a viscosity of perhaps 106
cm2/sec or more to be concerned about viscous effects. Often, a Reynolds
number of ~103 or more defines the onset of turbulence, although this does
not apply for all possible flows.
B. High Viscosity Convection.
The following analysis is motivated by experiments and specifically the
observation that convecting fluids develop most of their temperature
difference in boundary layers where vertical motions are inhibited and so the
heat must be carried by conduction.
Consider a very simple system: a hot layer of fluid that is neutrally stable
(adiabatic) except that the upper surface is suddenly cooled by an amount
ΔT and kept at that new lower temperature. The fluid is motionless, and
simple diffusion ensues. The uppermost part of the fluid develops a cold
layer of thickness δ ≈ (κt)1/2, where t is the elapsed time.
Now although our Rayleigh-Benard analysis pertained to a layer with a
uniform temperature gradient, it should be apparent from the physics
underlying the instability that we can define a local Rayleigh number in
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which the lengthscale used is just that within which most of the temperature
drop occurs:
Ra = gαΔT.δ3/νκ
(14.13)
Evidently, a time will be reached when Ra reaches about 1000. At about this
time, a convective instability will grow faster than the rate at which the layer
is thickening and this surface layer will peel away, to be replaced by deeper,
hot fluid. This process can be repeated indefinitely.
The time tconv between peelings is given by:
103 ≈ gαΔT(κtconv)3/2/νκ
⇒ tconv ≈ 102.(ν/gαΔTκ1/2)2/3
(14.14)
Clearly, the mean heat flow is about kΔT/δ≈ 0.1kΔΤ4/3( gα/νκ)1/3. To
remove Earth’s heat flow we need δ ≈100km. The implied viscosity by the
above critical Rayleigh number argument is then given by
103~gαΔTδ3/νκ ⇒ ν∼ gαΔTδ3/103κ ∼(103)(3x10-5)(103)(107)3/(103)(10-2)
whence ν ~1021 to 1022 cm2/sec, rather similar to the value inferred from
postglacial rebound for the mean mantle*. The value of tconv evaluated at
δ=100km is a couple of hundred million years. If there were patches of the
surface of scale L ~ few thousand km, then this implies a velocity for
material being peeled away of (a few thousand km)/(a few hundred million
years).....or a few cm per year.
*Postglacical rebound is the slow viscous response of earth’s surface
following deglaciation. It has the characteristic timescale we found earlier
for simple viscous flow arising from a Rayleigh–Taylor instabilty. The only
difference is that it’s not an instability: It is merely the return of the fluid to
its lowest energy state, driven by gravity. The postglacial response has (by
coincidence) occurred on a timescale ~ the time since the last ice age, i.e.,
~104 years, whence the viscosity estimate mentioned.
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_____________________________________________________________
This simple calculation shows something extraordinary: we can get plausible
(i.e. earth-like) values of heat flow for the values of viscosity inferred from
postglacial rebound using a model which requires no information about the
depth of the fluid (other than the requirement that it be much deeper than δ).
This is the fundamental characteristic of convection: it introduces a new
length scale (the thermal boundary layer thickness) which is largely
determined by the heat flow and material properties and not (directly or
strongly) by the geometry or size of the system.
Actually, as we shall see, the success of this model may be somewhat
fortuitous because Earth has plate tectonics, and this imperfectly understood
phenomenon is not the same as the above simple picture of convection.
Problems
14.1 A lot of laboratory and theoretical work is concerned with the Boussinesq
approximation, in which the only density variation permitted in the equations is that
arising from thermal expansion. The adiabatic temperature gradient and background
density variation (due to pressure gradients) are omitted. This has sometimes led to doubt
about the relevance of applying Boussinesq results using just the superadiabatic
temperature difference. In the case of low viscosity systems, the ΔT available to drive the
convection may be many orders of magnitude less than the actual ΔT. Is there any
justification for the Boussinesq approach in this situation? Can you devise an example
system that illustrates your conclusion?
Solution: The ΔTna (where “na” means non-adiabatic) is the quantity that enters into the
Rayleigh number. If we imagine a system that has a supercritical Rayleigh number, then
for a given value of this parameter, ΔTna ∝ L-3. The adiabatic part of the total temperature
drop is ΔTa ∝ pressure drop ∝ L . The regime of interest is one where both ΔTna and ΔTa
are small but ΔTna << ΔTa. In that regime, the variation in density due to pressure, and
other non-Boussinesq effects are small so the Boussinesq theory will be a good
approximation despite the smallness of ΔTna . Here is a concrete example: Suppose we
have ΔTna = 10-6 K, and earth-core-like gravity and other parameters. Then Ra =1020
(L/2000km)3. For ΔTa =(1000K).(L/2000km), the fractional density variation is ~
0.25.(L/2000km) (based on a Gruneisen gamma of ~1). In this situation, we would have
ΔTna << ΔTa <<T for 20m <L<~200km (and yet still have vigorous convection at the
small end for L, marginally accessible in the lab!) In summary, there is a very wide range
of conditions for which Boussinesq is an excellent approximation even though the actual
temperature drop is large compared to the part that drives convection. This range does not
(however) extend all the way to Earth’s core.
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14.2 Our analysis of the vorticity equation led to the equation (σ+ νk2) (σ+ κk2) = Agαβ
(where for simplicity the difference between the various component of the vector k are
ignored and simply replaced by the constant coefficient A on the RHS). Setting σ=0
defines a critical Rayleigh number Rac= (2π)4/A assuming k= 2π/L. But it is of interest to
examine the rapidity with which disturbances grow as a function of the parameter Ra/Rac.
Find σ in the form σ /κk2= f(Ra/Rac, Pr) where f is some specified simple function and
the Prandtl number Pr = ν/κ. Sketch this function in log–log space (i.e., covering Ra/Rac
all the way from near unity to ~1010 and covering Pr from 10-6 to 1030) and identify the
different asymptotic regimes where the function is particularly simple. In the case of
Earth’s mantle (Pr= 1024 , κ =10-2 cm2/sec, L=3000km), for what value of Ra/Rac is the
instability timescale σ-1 about ten million years?
Commentary: The actual Ra/Rac is typically estimated to be ~104 or 5. Your result (if
correctly done) is considerably smaller but it assumes that the temperature excess driving
convection is spatially uniform. In reality, the real system concentrates most of the
temperature anomaly into boundary layers. As a result, the mantle is not as unstable as
this simple calculation might suggest.
14.3 Immediately after the giant impact that led to the formation of the Moon, Earth was
probably radiating at 2500K and therefore had a heat flow corresponding to a black body
of this temperature. Assuming that this heat flow is accommodated by cooling from a
deep magma ocean, depth 3000km, estimate the convective overturn time for this ocean.
Common sense (guided by information earlier in this text) should enable you to make
good guesses of the parameters that you need, but for definiteness use Cp =107 erg/g.K,
α=10-4 K-1 and assume viscosity does not matter. How large would viscosity need to be
in order that it have a significant effect on your result? Does thermal diffusivity matter?
[If this entire problem takes more than half a page then you are doing something
unnecessary.]
Comment: Although silicate melts can be polymeric and therefore have large viscosities
(e.g., up to 103 cm2/sec for basaltic melt), the very high temperature magma implied by
this problem will likely have a viscosity more like that of water, i.e., 10-2 cm2/s. The
numbers quoted here are of course for kinematic viscosity, usually denoted ν.