Oceanography 540--Marine Geological Processes--Autumn Quarter 2002

Convection and Rayleigh Criteria

To motivate the Rayleigh number, we consider a scaling argument from ((13), p. 208 ff.). Imagine two horizontal plates, separated by a distance h, with increasing temperature downward such that the temperature difference between them is DeltaTh. Consider a parcel of fluid of scale d (d x d x d) located between these plates.

Rayleigh scaling

Figure 8-1

If displaced upwards it will have a buoyancy force acting on it, i.e., its density will be different from its surroundings. The buoyancy force is:

Eq 8-2: eq 9-1

where g is gravitational acceleration and rho is density. Further suppose that this density contrast is solely due to a contrast between its temperature and the surroundings:

Eq 8-3: eq 9-2

where alpha is the thermal expansion coefficient and T is temperature. Because the parcel is less dense than its surroundings, it will tend to move upwards. A viscous force:

Eq 8-4: eq 9-3

will act against the buoyancy force. In this expression µ is viscosity, z is the vertical spatial coordinate, and t is time. When these two forces balance:

Eq 8-5: eq 9-4

In addition, the parcel will be losing heat to its new surroundings by conduction, thereby reducing its temperature. The rate at which heat is lost will be proportional to the surface area of the parcel. Thus temperature (and so the density term) is a function of time:

Eq 8-6: eq 9-5

where kappa is the thermal conductivity. Combining 8-3, 8-5 and 8-6 and rearranging:

Eq 8-7: eq 9-6

How far can the parcel move in infinite time? We integrate from zero to infinity:

Eq 8-8: eq 9-7

to find that the distance traversed, z, is:

Eq 8-9: eq 9-8

The terms in the numerator of equation 8-9 have their origin in the buoyancy force and promote upward motion while increasing either viscosity or thermal conductivity in the denominator limits upward motion.

To be able to convect heat, there must be movement across the distance between the two plates separated by a distance h in finite time and so:

Eq 8-10: eq 9-9

By scaling the fluid parcel to the plate separation, through an arbitrary factor f:

Eq 8-11: eq 9-10

substituting into equation 8-10 and rearranging it follows that:

Eq 8-12: eq 9-11

The left hand side of equation 8-12 is called the Rayleigh number:

Eq 8-13: eq 9-12

Lord Rayleigh showed through linear perturbation analysis (for a general discussion see (13), section 7.1.2 or the derivation in (11), section 6-18), that for thermal instabilities to grow in a fluid, the Rayleigh number would have to exceed a critical value (the f^4 scale on the right hand side). When:

Eq 8-14: eq 9-13

the temperature profiles between the two plates will be conductive, while when:

Eq 8-15: eq 9-14

a cellular convection will be established, displacing the isotherms upward in the zone of rising fluid and downward in the zone of downwelling. The value of the critical Rayleigh number depends on the particular geometry and boundary conditions. For convection between parallel plates it lies between 657 (free boundaries) and 1708 (rigid boundaries).

This example is for water convecting between two rigid boundaries for Ra~3000. The temperature field exhibits two narrow boundary layers, with convective stirring between. This temperature distribution is reflected in the velocity field, with sluggish flow at the boundaries and in the interiors of the two cells, strong downwelling at the side boundaries, and strong upwelling in the middle of the domain:

temperature field

Figure 8-2. Temperature field for a system with Ra~3000.

velocity field

Figure 8-3. Velocity field for a system with Ra~3000.

Nusselt Number

The Nusselt number is the ratio of heat transferred from a convecting system compared to that which would be transferred by conduction alone. Thus, below the critical Rayleigh number, the Nusselt number is one. Once the critical Rayleigh number is exceeded, it grows as (Ra/Ra,crit)^1/3. The Nusselt number is computed by integrating the heat flux through the upper surface of the convecting system.

nusselt number near the critical Rayleigh
number Figure 8-4. Nusselt number near the critical Rayleigh number.

Will fluids in the ocean crust convect?

Consider a open fracture penetrating to great depth, say a kilometer. The Rayleigh number is:

Eq 8-16: eq 9-15

(Here the viscosity and density term have been replaced with the kinematic viscosity.). For some representative values

Table](Table VII-1)

the Rayleigh number is about 10^2^3--far in excess of the critical value.


Animation of Unsteady 2-D Convection

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