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Q1 The Rayleigh and Nusselt number Recall that the Nusselt number is given by \[ N u=\frac{R_{m}}{S_{e}\left(T_{\mathrm{CMB}}-T_{0}\right)} \int_{S}\left(\frac{\partial T}{\partial z}\right)_{z=0} d S \] where \( R_{m} \) represents the thickness of the mantle, \( S_{e} \) represent the surface area of the Earth, \( T_{\mathrm{CMB}} \) is the temperature at the core-mantle boundary and \( T_{0} \) is the surface temperature. For the following calculations, you can assume that the temperature at the CMB is 4500 K and that the mantle is entirely composed of dry Olivine with a conductivity given by \( k=1.67 \mathrm{~W} \mathrm{~m}^{-1} \mathrm{~K}^{-1} \) and a thermal expansivity of \( \alpha=3 \times 10^{-5} \mathrm{~K}^{-1} \). (a) Derive an expression for the Rayleigh number (Ra) for a Boussinesq fluid convecting in a box of height \( h \). Assume that the fluid is incompressible, constant viscosity, there are no internal sources of heating and that the thermal conductivity is constant. A fixed temperature boundary condition is applied on the top and the bottom of the domain. Use the following choices for the non-dimensional scaling: \[ \boldsymbol{x}^{\prime}=\boldsymbol{x} / h, \quad t^{\prime}=t /\left(h^{2} / \kappa\right), \quad T^{\prime}=T / \Delta T, \quad \boldsymbol{v}^{\prime}=\boldsymbol{v} /(\kappa / h), \quad p^{\prime}=p /\left(\eta \kappa / h^{2}\right), \] where \( h \) and \( \Delta T \) are the characteristic length (height of the box) and \( \Delta T \) is the temperature difference between the lower and upper boundary. When deriving \( R a \), make sure you consider the perturbed form of the momentum equations.

          Q1 The Rayleigh and Nusselt number
Recall that the Nusselt number is given by
\[
N u=\frac{R_{m}}{S_{e}\left(T_{\mathrm{CMB}}-T_{0}\right)} \int_{S}\left(\frac{\partial T}{\partial z}\right)_{z=0} d S
\]
where \( R_{m} \) represents the thickness of the mantle, \( S_{e} \) represent the surface area of the Earth, \( T_{\mathrm{CMB}} \) is the temperature at the core-mantle boundary and \( T_{0} \) is the surface temperature. For the following calculations, you can assume that the temperature at the CMB is 4500 K and that the mantle is entirely composed of dry Olivine with a conductivity given by \( k=1.67 \mathrm{~W} \mathrm{~m}^{-1} \mathrm{~K}^{-1} \) and a thermal expansivity of \( \alpha=3 \times 10^{-5} \mathrm{~K}^{-1} \).
(a) Derive an expression for the Rayleigh number (Ra) for a Boussinesq fluid convecting in a box of height \( h \). Assume that the fluid is incompressible, constant viscosity, there are no internal sources of heating and that the thermal conductivity is constant. A fixed temperature boundary condition is applied on the top and the bottom of the domain. Use the following choices for the non-dimensional scaling:
\[
\boldsymbol{x}^{\prime}=\boldsymbol{x} / h, \quad t^{\prime}=t /\left(h^{2} / \kappa\right), \quad T^{\prime}=T / \Delta T, \quad \boldsymbol{v}^{\prime}=\boldsymbol{v} /(\kappa / h), \quad p^{\prime}=p /\left(\eta \kappa / h^{2}\right),
\]
where \( h \) and \( \Delta T \) are the characteristic length (height of the box) and \( \Delta T \) is the temperature difference between the lower and upper boundary. When deriving \( R a \), make sure you consider the perturbed form of the momentum equations.

Q1 The Rayleigh and Nusselt number
Recall that the Nusselt number is given by

N u=(Rm)/(Se(TCMB-T0))∫S((∂ T)/(∂ z))z=0 d S

where Rm represents the thickness of the mantle, Se represent the surface area of the Earth, TCMB is the temperature at the core-mantle boundary and T0 is the surface temperature. For the following calculations, you can assume that the temperature at the CMB is 4500 K and that the mantle is entirely composed of dry Olivine with a conductivity given by k=1.67 W m^-1 K^-1 and a thermal expansivity of α=3 × 10^-5 K^-1.
(a) Derive an expression for the Rayleigh number (Ra) for a Boussinesq fluid convecting in a box of height h. Assume that the fluid is incompressible, constant viscosity, there are no internal sources of heating and that the thermal conductivity is constant. A fixed temperature boundary condition is applied on the top and the bottom of the domain. Use the following choices for the non-dimensional scaling:

x^'=x / h, t^'=t /(h^2 / κ), T^'=T / Δ T, v^'=v /(κ / h), p^'=p /(ηκ / h^2),

where h and Δ T are the characteristic length (height of the box) and Δ T is the temperature difference between the lower and upper boundary. When deriving R a, make sure you consider the perturbed form of the momentum equations.

Added by Darnell C.

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