Reescritura en cartesianas de la reformulación covariante del sector elíptico de la aproximación CFC en términos de CoCoNuT

Ya escribimos al respecto en este post. Aquí lo que haremos es reescribir las expresiones allí introducidas

En primer lugar, teniamos:

 $latex Delta X^i = 8 pi f^{ij}S_j^* – frac{1}{3}mathcal{D}^i mathcal{D}_j X^j$

donde:

$latex S_j^* := sqrt{ frac{gamma}{f} } S = psi^6 S_j$,

$latex S_j := rho h w^2 v_j$.

En el caso de estar trabajando en cartesianas y teniendo en cuenta todo el trabajo realizado en el artículo, nos queda:

$latex partial_{xx} X^x + partial_{yy} X^x + partial_{zz} X^x = 8 pi psi^6 rho h w^2 v_x – frac{1}{3} partial_x (partial_x X^x + partial_y X^y + partial_z X^z)$,

$latex partial_{xx} X^y + partial_{yy} X^y + partial_{zz} X^y = 8 pi psi^6 rho h w^2 v_y – frac{1}{3} partial_y (partial_x X^x + partial_y X^y + partial_z X^z)$,

$latex partial_{xx} X^z + partial_{yy} X^z + partial_{zz} X^z = 8 pi psi^6 rho h w^2 v_z – frac{1}{3} partial_z (partial_x X^x + partial_y X^y + partial_z X^z)$.

A continuación, y para la siguiente ecuación, necesitamos:

$latex hat{A}^{ij} = mathcal{D}^i X^j + mathcal{D}^j X^i – frac{2}{3} mathcal{D}_k X^k f^{ij}$

que queda como:

$latex hat{A}^{xx} = 2 partial_x X^x – frac{2}{3} (partial_x X^x + partial_y X^y + partial_z X^z)$,

$latex hat{A}^{xy} = hat{A}^{yx}= partial_x X^y + partial_y X^x$,

$latex hat{A}^{xz} = hat{A}^{zx} = partial_x X^z + partial_z X^x$,

$latex hat{A}^{yy} = 2 partial_y X^y – frac{2}{3} (partial_x X^x + partial_y X^y + partial_z X^z)$,

$latex hat{A}^{yz} = hat{A}^{zy} = partial_y X^z + partial_z X^y$,

$latex hat{A}^{zz} = 2 partial_z X^z – frac{2}{3} (partial_x X^x + partial_y X^y + partial_z X^z)$,

por lo que:

$latex Delta psi = -2 pi psi^{-1} E^* – psi^{-7} frac{f_{il}f_{jm}hat{A}^{lm}hat{A}^{ij}}{8}$

donde:

$latex E^*:= sqrt{ frac{gamma}{f} } E = psi^6 E$,

$latex E:= D + tau$

es:

$latex Delta psi = -2 pi psi^{-1} (D + tau) – psi^{-7} frac{(hat{A}^{xx})^2+(hat{A}^{yy})^2+(hat{A}^{zz})^2+2(hat{A}^{xy})^2+2(hat{A}^{xz})^2+2(hat{A}^{yz})^2}{8}$.

La siguiente:

$latex Delta (alphapsi) = 2 pi (alphapsi)^{-1} (E^* + 2S^*) + frac{7}{8} (alphapsi)^{-7} (f_{il} f{jm} hat{A}^{lm} hat{A}^{ij})$

con:

$latex S^*:= sqrt{ frac{gamma}{f} } S = psi^6 S$,

$latex S:= rho h (w^2-1) + 3 p$

queda:

$latex Delta (alphapsi) = 2 pi (alphapsi)^{-1} ( D + tau + 2 rho h (w^2-1) + 6 p) + $

$latex + frac{7}{8}(alphapsi)^{-7} ((hat{A}^{xx})^2+(hat{A}^{yy})^2+(hat{A}^{zz})^2+2(hat{A}^{xy})^2+2(hat{A}^{xz})^2+2(hat{A}^{yz})^2)$

Y la última:

$latex Delta beta^i = mathcal{D}_j (2 (alphapsi)^{-6} hat{A}^{ij}) – frac{1}{3} mathcal{D}^i (mathcal{D}_j beta^j)$,

que escribimos como:

$latex Delta beta^x = partial_x (2 (alpha psi)^{-6} hat{A}^{xx}) + partial_y (2 (alpha psi)^{-6} hat{A}^{xy}) + partial_z (2 (alpha psi)^{-6} hat{A}^{xz}) – $

$latex – frac{1}{3} partial_x (partial_x beta^x + partial_y beta^y + partial_z beta^z)$

$latex Delta beta^y = partial_x (2 (alpha psi)^{-6} hat{A}^{yx}) + partial_y (2 (alpha psi)^{-6} hat{A}^{yy}) + partial_z (2 (alpha psi)^{-6} hat{A}^{yz}) – $

$latex – frac{1}{3} partial_y (partial_x beta^x + partial_y beta^y + partial_z beta^z)$

$latex Delta beta^z = partial_x (2 (alpha psi)^{-6} hat{A}^{zx}) + partial_y (2 (alpha psi)^{-6} hat{A}^{zy}) + partial_z (2 (alpha psi)^{-6} hat{A}^{zz}) – $

$latex – frac{1}{3} partial_z (partial_x beta^x + partial_y beta^y + partial_z beta^z)$