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Further admitting that lateral inhomogeneity with the study area is not high the Thermal Conductivity is going to be a function of depth only Thermal Conductivityof true vertical depth only | LaTeX Math Inline |
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| body | \lambda_r(x,y,z) = \lambda_r(z) |
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which leads to vanishing lateral components of the heat flux: | LaTeX Math Inline |
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| body | --uriencoded--%7B \bf j%7D(x,y,z) = \%7B j_x = 0, \, j_y = 0 , \, j_z(z) \%7D |
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| body | --uriencoded--%7B \bf j%7D(x,y,z) = \%7B 0, \, 0 , \, j_z = %7B\rm const%7D \%7D |
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| body | \lambda_r(x,y,z) = \lambda_r(z) |
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Equations
| Neutral Layer |
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| z_n = z_s + H_n |
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| H_n = \sqrt{\frac{a_e \, A_T }{\pi}} \, \ln \frac{\delta T_A }{\delta T_{\rm cut} } |
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| Below Neutral Temperature Layer | Above Neutral Temperature Layer |
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| T_G(l) = T_s + \int_{z_s}^z G_T(z) dz |
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| T(t, z) = T_0 + \frac{j_z}{\lambda_er(z)} (z-z_s) + T_Y(t, z) + T_D(t, z) |
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| G_T(z) =\frac{d T_g}{d z}= \frac{j_z}{\lambda_r} |
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| T_Y(t,z) = \delta T_A \, \exp \left[ \, {(z_s-z}) \sqrt{\frac{\pi}{a_e \, A_T}} \, \right] \, \cos \left[ \, 2 \pi \frac{t - \delta t_A}{A_T} + (z_s -z) \sqrt {\frac{\pi}{a_e \, A_T}} \, \right] |
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| T_D(t,z) = \delta T_D \, \exp \left[ \, {(z_s-z}) \sqrt{\frac{\pi}{a_e \, D_T}} \, \right] \, \cos \left[ \, 2 \pi \frac{t - \delta t_D}{D_T} + (z_s -z) \sqrt {\frac{\pi}{a_e \, D_T}} \, \right] |
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