potentialForRiverbedChange.ipynb

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   "source": [
    "#  Simplified method for predicting the potential for riverbed change"
   ]
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   "source": [
    "## 導入\n",
    "\n",
    " - 摩擦速度や無次元掃流力を用いることが多い。これらは、河床土砂の動きやすさを示すものであり、河床変動を生じやすさを直接示すものではない。\n",
    " - 流れの計算のみ簡易予測\n",
    " - ALB等面的河床地形情報の取得により流れの計算精度は向上、一方で、河床変動は粒度分布のデータの不足等により・・・精度が低い。"
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   "metadata": {},
   "source": [
    "## 基礎式の導出\n",
    "\n",
    "平面二次元流（デカルト座標系）における掃流砂による単位時間あたりの河床変動高は、河床の連続式（質量保存則）より次式で示される。\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "    \\dfrac{\\partial z_b}{\\partial t} &= -\\dfrac{1}{1-\\lambda} \\left(\\dfrac{\\partial q_b}{\\partial x} + \\dfrac{\\partial q_b}{\\partial y} \\right) \n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "単一粒径場を想定すると、流砂量の一般形は次式で示される。\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "    \\dfrac{q_b}{\\sqrt{\\rho g d^3}} &= \\alpha \\left(\\dfrac{u_*^2}{\\rho g d}\\right)^{\\beta}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "ここに、$\\alpha、\\beta$は各種流砂量式で設定される係数である。$\\beta$は、国内でよく使用される芦田・道上式、MPM式では$3/2$である。よって、\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "    q_b =  \\dfrac{\\alpha u_*^3}{\\rho g }\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "である。\n",
    "\n",
    "流砂量式を河床の連続式に代入し整理すると、次式が導出される。\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "    \\dfrac{\\partial z_b}{\\partial t} &= -\\dfrac{\\alpha}{(1-\\lambda)\\rho g} \\left(\\dfrac{\\partial u_{*x}^3}{\\partial x} + \\dfrac{\\partial u_{*y}^3}{\\partial y} \\right) \n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "$\\alpha,\\lambda,\\rho,g$は定数であるため、下式の比例関係が示される。ここで、右辺を河床変動ポテンシャルと定義する。\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\dfrac{\\partial z_b}{\\partial t} \\propto  -\\dfrac{\\partial u_{*x}^3}{\\partial x} -\\dfrac{\\partial u_{*y}^3}{\\partial y}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "又はこっちの方がいいかな？\n",
    "$$\n",
    "\\begin{align}\n",
    "\\dfrac{\\partial z_b}{\\partial t}  =   a \\left(- \\dfrac{\\partial u_{*x}^3}{\\partial x} -\\dfrac{\\partial u_{*y}^3}{\\partial y}\\right)\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "ここに、$a$は正の実数"
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   "source": [
    "## 一般座標系の展開\n",
    "\n",
    "\n",
    "本検討では一般座標系の平面二次元計算を行うため、上式は次のとおりになる。\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "(1-\\lambda)\\dfrac{\\partial}{\\partial t }\\left(\\dfrac{z_b}{J}\\right) &= -\\left[ \\dfrac{\\partial}{\\partial \\xi}\\left( \\dfrac{ q_{b\\xi}}{J} \\right) +  \\dfrac{\\partial}{\\partial \\eta}\\left( \\dfrac{ q_{b\\eta}}{J} \\right) \\right] \\\\\n",
    "J &= \\dfrac{1}{ \\dfrac{\\partial x }{\\partial \\xi} \\dfrac{\\partial y }{\\partial \\eta} + \\dfrac{\\partial x }{\\partial \\eta} \\dfrac{\\partial y }{\\partial \\xi}} \\\\\n",
    "q_{b\\xi} &= \\dfrac{\\partial \\xi }{\\partial x} q_{bx} + \\dfrac{\\partial \\xi }{\\partial y} q_{by} \\\\\n",
    "q_{b\\eta} &= \\dfrac{\\partial \\eta }{\\partial x} q_{bx} + \\dfrac{\\partial \\eta }{\\partial y} q_{by}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\dfrac{\\partial z_b}{\\partial t} &\\propto  -\\dfrac{\\partial u_{*\\xi}^3}{\\partial \\xi} -\\dfrac{\\partial u_{*\\eta}^3}{\\partial \\eta} \\\\\n",
    "u_{*\\xi} &= \\dfrac{\\partial \\xi }{\\partial x} u_{*x} + \\dfrac{\\partial \\xi }{\\partial y} u_{*y} \\\\\n",
    "u_{*\\eta} &= \\dfrac{\\partial \\eta }{\\partial x} u_{*x} + \\dfrac{\\partial \\eta }{\\partial y} u_{*y}\n",
    "\\end{align}\n",
    "$$\n"
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