Temperature evolution of an oceanic fracture one Xiaohua Xu & Zhao Chen October 29, 212
Outline Background Introduction to the Problem Green's Function and the Analytical Solution for Temperature Introduction to Green s Function Solution of Temperature by method of Green s Function Model Topography V.S. Real Topography Solution the Model Topography Topography map with the evolution of Temperature Comparison with the real Topography Conclusions and Discussions
1 Background
1 Background
1 Background The Equation of Heat Conduction: Initial Condition: Boundary Condition: 2 2 1 2 2 x t T T T T( x,, t T ( x m T( x,, t Tmerf ( ( x 2 t T( x,, t = T( x,, t Tm x Automatically Satisfied
2.1 Green's Function The Green s Function can be treated as the temperature at (x,y, at the time t, due to an instantaneous point source of strength unity generated at the point P(x,y, at time t. We solve G and make a Convolution with the Initial Condition T, we will get the solution. 2 2 G ( G G ( x x 2 2 ( ( t t t x The Green s Function also satisfy the same B.C. as T does. G( x,, t = To satisfy this boundary condition, we can make an analogy with what we ve learned in Electromagnetics, which means that we need a Imaginary source that absorbs heat at the mirror point of the original heat source.
2.1 Green's Function -q -Q U= x T= x E Heat Flow q Q The equilibrium function of U and T are all Laplace s Equation. To satisfy T= at =, what we need is to make the final solution of the Green s Function to be G(- -G(+ (since the upper source is negative.
2.2 Solution to the Green s Function 2 2 G ( G G ( x x ( ( t t 2 2 t x Take a Fourier Transform to Green s Function. G 2 2 2 2 i ( kxxk 4 ( k x k G+e ( t t t Apply the formula for the solution of First Order ODE. For y' P( x y Q( x, the solution will be: So, G x x y e Q( x e dx t t 2 2 2 2 2 2 x x t 2 i( kxx k t 4 ( k k dt 4 ( k k dt e e ( t t e dt e x P( t dt P( t dt 2 2 2 kx k t t i kxx k e 4 ( ( 2 ( x
2.2 Solution to the Green s Function Take a Inverse Fourier Transform to the solution. 2 2 2 4 ( kx k ( tt 2 i( kxx k 2 i( kxxk G e e e dkxdk 1 2 4 ( t t i( xx i( [2 ( tt k ] [2 ( tt k ] e 2 2 x 4 ( tt 4 ( tt 2 2 ( xx ( 4 ( tt e d(2 ( t t k d(2 ( t t x k 1 4 ( t t e 2 2 ( xx ( 4 ( tt Finally, 2 2 2 2 ( xx ( ( xx ( 4 ( tt 4 ( tt ( 1 G e e 4 ( t t
2.3 Solution to Temperature Evolution The Initial Condition of T is T '( x,, t (1 H( x Tm TmH ( x erf ( 2 t The Temperature will be the Convolution of T and G T ( x,, t T '( I. C.* G T '( x,, t G( x x,, t t dx d ( TmGdx Tmerf ( Gdx d 2 t x x 1, 2, 3 4 ( t t 4 ( t t 4 ( t t
2.3 Solution to Temperature Evolution 2 2 ( xx ( 2 2 ( xx ( ( 4 ( tt 4 ( tt 1 T Gdx d T e e dx d m m 4 ( t t T m 4 T 4 T 4 m m 2 ( 2 ( erfc( x 2 4 ( tt 4 ( tt x 2 4 ( tt 4 ( tt 2 2 1 2 e e d d 2 1 2 3 1 2 e e d d x erfc( 4 ( t t 4 ( t t Tm x erfc( erfc( 4 4 ( t t 4 ( t t Tm x erfc( erf ( 2 4 ( t t 4 ( t t 1 3 Solution to the first part
2.3 Solution to Temperature Evolution T erf ( Gdx d m 2 t 2 2 ( ( x 2 1 4 ( tt 4 ( tt ( ( ( Tm erfc 4 e e erf d 4 ( t t 4 ( t t 2 t Consider the first part: erf ( * G 2 t 2 1 1 = F F erf G 2 ( 4 ( tt e erf ( d 4 ( t t 2 t [ ( ( ] 2 t Fourier Transform just come to my mind
2.3 Solution to Temperature Evolution 1 1 2 F F F 2 t 2ik 2 t 2ik 2 ( erf ( ( erf '( ( ( 2 t e d / 1 1 1 1 F ( 2ik t 2ik t 2 2 ( ( 2 t 2 t 2ik e e e d 1 2 t 2 ik t 2 t 1 2 2 4 tk e i k 1 erf G e e 2 t i k 2 2 2 ( 2 i tk 4 tk 2 t e d i t k 2 2 2 2 tk t t k 1 4 4 ( ( * F [ ] F 1 i k 2 2 tk 1 4 [ e ] erf ( 2 t ( 2
2.3 Solution to Temperature Evolution Source Imginary Source 2 tt ( 4 ( 2 1 e erf ( d erf ( 4 ( t t 2 t 2 t 2 tt ( 4 ( 2 1 e erf ( d erf ( 4 ( t t 2 t 2 t T erf ( Gdx d m 2 t 2 2 ( ( x 2 1 4 ( tt 4 ( tt ( ( ( Tm erfc e e erf d 4 4 ( t t 4 ( t t 2 t Tm x erfc( erf ( 2 4 ( t t 2 t Solution to the second part
2.3 Solution to Temperature Evolution T ( x,, t ( T Gdx T erf ( Gdx d m m 2 t Tm x [ erfc( erf ( 2 4 ( t t 4 ( t t x erfc( erf ( ] 4 ( t t 2 t
3.1 Bathymetry: Solution of Model Topography 2D model for Ocean Floor Topography by Isostasy. (P175 w 2 ( T T t m v m m w 1/2 3D model for Ocean Floor Topography by Isostasy: Plug Temperature to the model and do the Integral on P175 1/2 m vtm t x t x ( t t w erfc( erfc( m w 2 t 2 t Arbitrarily defination: v m 5 1 6 2 1 3 1 K, 1 m s, Tm 13 K, kg m kg m t Myr 3 3 33, w 1, 2
3.2 Model Temperature and Topography t, t 2 Myr, Temperature
3.2 Model Temperature and Topography t 1 Myr, t 2 Myr, Temperature
3.2 Model Temperature and Topography t 5 Myr, t 2 Myr, Temperature
3.2 the Model Temperature and Topography t 1 Myr, t 2 Myr, Temperature
3.2 Model Temperature and Topography t 2 Myr, t 2 Myr, Temperature
3.2 Model Temperature and Topography t 3 Myr, t 2 Myr, Temperature
3.2 Model Temperature and Topography t, t 2 Myr, Topography
3.2 Model Temperature and Topography t 1 Myr, t 2 Myr, Topography
3.2 Model Temperature and Topography t 5 Myr, t 2 Myr, Topography
3.2 Model Temperature and Topography t 1 Myr, t 2 Myr, Topography
3.2 Model Temperature and Topography t 2 Myr, t 2 Myr, Topography
3.2 Model Temperature and Topography t 3 Myr, t 2 Myr, Topography
3.3 Comparison with real Topography
3.3 Comparison with real Topography t 5 Myr, t 28 Myr, Topography Model Reality Long:129W Lat:36-42N Data all set as in the paper
3.3 Comparison with real Topography t 15 Myr, t 28 Myr, Topography Model Reality Long:133W Lat:36-42N
3.3 Comparison with real Topography t 25 Myr, t 28 Myr, Topography Model Reality Long:137W Lat:36-42N
3.3 Comparison with real Topography t 35 Myr, t 28 Myr, Topography Model Reality Long:141W Lat:36-42N
3.3 Comparison with real Topography t 45 Myr, t 28 Myr, Topography Model Reality Long:144W Lat:36-42N
4 Conclusions and Discussion (1.It works well for this model on Difference, but not quite well on actually depth of the topography. Top of the ridge? Sediments? (2.The model is smoothed but real topography is uneven and have many spikes on it. The model of Isostasy is too ideal. Neglected factors? Solution in the paper based on elastic model can explain the spikes to some extent. (3.About the initial condition. ERF function?
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