Temp budget calc

tcalc.ipynb

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    "## idea:\n",
    "\n",
    "Calculate the temperature flux is a region from $x=0$ to $x=L$, between $y=y_0$ and $y=y_0 + \\delta y$\n",
    "\n",
    "We divide the data into *three* components as in the paper. \n",
    "\n",
    "$$ T(x, y) = T_0(y) + T_x(y)\\,x + T'(x, y) $$\n",
    "\n",
    "$$ v(x, y) = v_0(y) + v_x(y)\\,x + v'(x, y) $$\n",
    "\n",
    "$$ u(x, y) = u_0(y) + u_x(y)\\,x + u'(x, y) $$\n",
    "\n",
    "The resolved flux out the \"north\" side is \n",
    "\n",
    "$$\\int_0^L vT \\ \\mathrm{d}x $$\n",
    "\n",
    "The decomposition has 9 terms, of which at least two integrate to zero:\n",
    "\n",
    "$$\\int_0^L vT \\ \\mathrm{d}x = v_0T_0 L + v_0 T_x \\frac{L^2}{2} + 0 + v_x T_0 \\frac{L^2}{2} + v_x T_x \\frac{L^3}{3} +  v_x \\int_0^L  T' x\\ \\mathrm{d}x + 0 +  T_x \\int_0^L v' x\\ \\mathrm{d}x + \\int_0^L v' T' \\mathrm{d}x   $$\n",
    " \n",
    "Given that, I don't understand how so many terms dropped out in the paper's eq 7.   You seem to have decided that whatever you are calling $\\Delta T  = T_x \\ x$ has no x-dependence.  If so, thats incorrect.   \n",
    "\n",
    "Eq 7 also has a mysterious $\\Delta \\Pi'_a$.  What is this?  \n",
    "\n",
    "\n",
    "\n"
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