Transcript Convection in a planetary body

Convection in a planetary body
Geosciences 519
Natalie D. Murray
April 2, 2002
Convection
Process of heat transfer ( from hotter to colder regions)
by the bulk motion of a fluid
•More efficient in heat transfer than conduction
•Needs:
•Temperature gradient
•Gravity
Temperature structure of the mantle–
superadiabatic temperature gradient due to
heating from beneath (from the core) and
radiogenic heat production
http://www.ldeo.columbia.edu/users/jcm/Topics3/Topics3.html
Buoyancy
Parcel – unit volume of a fluid
Adiabatic process – no exchange of heat with surroundings
Simple process :
•Parcel is heated from below
•Temperature increases causing density changes
•Density of parcel is less than that of surrounding material
•Parcel is more buoyant and will rise
•Since the temperature gradient is superadiabatic and the
parcel rises adiabatically, the parcel is warmer than the
surroundings and will continue to rise.
Navier-Strokes Equations
Momentum equation

 

Du
1  
 2 x u   p  g   2u (eqn 1)
Dt

Mass Conservation equation
Conservation of Energy
Hydrostatic Equation
where

1 D
   u  0 (eqn 2)
 Dt
dE  dQ  dW (eqn 3)
dp
 g (eqn 4)
dz
D d 
d
d
d
d
  u    u  v  w
Dt dt
dt
dx
dy
dz
Boussinesq Approximation
Density variations are ignored except when coupled
with gravity and give rise to buoyancy (gravitational
force)
Prandtl Number

Pr 

•Virtually infinite in the mantle
•Inertial forces are insignificant
•Convection depends on pressure, temperature and
viscosity
Forces opposing convection

Viscosity – opposes fluid flow (for the Mantle – about the same as
for steel 1E20 Pa s)

Thermal diffusivity - suppress the temperature fluctuation by
causing the rising plume of hot fluid to equilibriate with
surrounding fluid (weakens the buoyancy force)
Rayleigh Number
Ratio of buoyancy force to the viscous – diffusive force
Convection due to
gTd
superadiabatic
Ra 

temperature gradient
Critical Rayleigh Number – value that if exceeded convection is certain
Rayleigh Number for the mantle is super critical
3

Thermal expansion coeff – the more a fluid expands, the
more it’s density is lowered
Typical values for coefficients can be found on Lowrie pg 328 Table 6.2
gQD5
RaQ 
k
Convection due to radiogenic heat
Lowrie pg 328
Stability
http://www.seas.smu.edu/~arunn/html/convect/rbconvect/rbcon.html
Simple Convective Cell
http://ldeo.columbia.edu/users/jcm/Topics3/Topics3.html
Rayleigh-Benard Convection
•Simple model of convection
•Thermal convection – transfer of heat through a fluid
•Parcel will rise to the level of neutral buoyancy
•The hot layer will try to rise while the cold layer will try to
sink.
•Breaks up into convective cells
•In the form of rolls, hexagon cells, etc.
Rayleigh-Benard Convection
http://www.seas.smu.edu/~arunn/html/convect/rbconvect/rbcon.html
Rayleigh-Benard Convection
http://www.ldeo.columbia.edu/users/jcm/Topics3/Topics3.html
Convective Cells in the Mantle
http://geollab.jmu.edu/Ficher/plateTect/heathistory.html
Helpful Websites
http://scienceworld.wolfram.com
http://www.psigate.ac.uk/