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| 1 | +# Thin-layer Boussinesq modules |
| 2 | + |
| 3 | +```math |
| 4 | +\newcommand{\bcdot}{\boldsymbol \cdot} |
| 5 | +\newcommand{\bnabla}{\boldsymbol \nabla} |
| 6 | +\newcommand{\pnabla}{\bnabla_{\! \perp}} |
| 7 | +
|
| 8 | +\newcommand{\com}{\, ,} |
| 9 | +\newcommand{\per}{\, .} |
| 10 | +
|
| 11 | +\newcommand{\bu}{\boldsymbol u} |
| 12 | +\newcommand{\bU}{\boldsymbol U} |
| 13 | +\newcommand{\buu}{{\boldsymbol{u}}} |
| 14 | +\newcommand{\bb}{{b}} |
| 15 | +\newcommand{\pp}{{p}} |
| 16 | +\newcommand{\ww}{{w}} |
| 17 | +\newcommand{\uu}{{u}} |
| 18 | +\newcommand{\v}{\upsilon} |
| 19 | +\newcommand{\vv}{{\upsilon}} |
| 20 | +\newcommand{\zzeta}{{\zeta}} |
| 21 | +\newcommand{\oomega}{{\omega}} |
| 22 | +\newcommand{\boomega}{\boldsymbol{\oomega}} |
| 23 | +
|
| 24 | +\newcommand{\bxh}{\widehat{\boldsymbol{x}}} |
| 25 | +\newcommand{\byh}{\widehat{\boldsymbol{y}}} |
| 26 | +\newcommand{\bzh}{\widehat{\boldsymbol{z}}} |
| 27 | +\newcommand{\ii}{\mathrm{i}} |
| 28 | +\newcommand{\ee}{\mathrm{e}} |
| 29 | +\newcommand{\cc}{\mathrm{c.c.}} |
| 30 | +\newcommand{\J}{\mathsf{J}} |
| 31 | +
|
| 32 | +\newcommand{\p}{\partial} |
| 33 | +``` |
| 34 | + |
| 35 | +These modules solve various thin-layer approximations to the hydrostatic Boussinesq |
| 36 | +equations. A thin-layer approximation is one that is appropriate for dynamics with |
| 37 | +small aspect ratios, or small vertical scales and large horizontal scales. |
| 38 | +Thin layer approximations include the shallow-water system, layered system, and |
| 39 | +spectral approximations that apply a Fourier or Sin/Cos eigenfunction expansion in the |
| 40 | +vertical coordinate to the Boussinesq equations, and truncate the expansion at just |
| 41 | +two or three modes. Approximations of this last flavor are described here. |
| 42 | + |
| 43 | +The three-dimensional rotating, stratified, hydrostatic Boussinesq equations are |
| 44 | + |
| 45 | +```math |
| 46 | +\p_t\buu + \left ( \buu \bcdot \bnabla \right ) \buu + f \bzh \times \buu + \bnabla \pp = D^{\buu} \com \\ |
| 47 | +\p_z \pp = \bb \com \\ |
| 48 | +\p_t\bb + \ww N^2 = D^\bb \com \\ |
| 49 | +\bnabla \bcdot \buu = 0 \com |
| 50 | +``` |
| 51 | + |
| 52 | +where $\bu = (u, \v, w)$ is the three-dimensional velocity, $b$ is buoyancy, $p$ is pressure, $N^2$ is the |
| 53 | +buoyancy frequency (constant), and $f$ is the rotation or Coriolis frequency. The operators $D^{\buu}$ and $D^{\bb}$ are arbitrary dissipation that we define only after projecting onto vertical Fourier or Sin/Cos modes. |
| 54 | +Taking the curl of the horizontal momentum equation yields an evolution |
| 55 | +equation for vertical vorticity, $\zzeta = \p_x \vv - \p_y \uu$: |
| 56 | + |
| 57 | +```math |
| 58 | +\p_t\zzeta + \buu \bcdot \bnabla \zzeta - \left (f \bzh + \boomega \right ) |
| 59 | + \bcdot \bnabla \ww = D^{\zzeta} \per |
| 60 | +``` |
| 61 | + |
| 62 | +## Vertically Fourier Boussinesq |
| 63 | + |
| 64 | +The vertically-Fourier Boussinesq module solves the Boussinesq system obtained by expanding the hydrostatic |
| 65 | +Boussinesq equations in a Fourier series. The horizontal velocity $\uu$, for example, is expanded with |
| 66 | + |
| 67 | +```math |
| 68 | +\uu(x, y, z, t) \mapsto U(x, y, t) + \ee^{\ii m z} u(x, y, t) + \ee^{-\ii m z} u^*(x, y, t) \com |
| 69 | +``` |
| 70 | + |
| 71 | +The other variables $\vv$, $\bb$, $\pp$, $\zzeta$, and $\boomega$ are expanded identically. The barotropic |
| 72 | +horizontal velocity is $V$ and the barotropic vertical vorticity is $Z = \p_x V - \p_y U$. The barotropic |
| 73 | +vorticity obeys |
| 74 | +```math |
| 75 | +\p_t Z + \J \left ( \Psi, Z \right ) |
| 76 | + + \bnabla \bcdot \left ( \bu \zeta^* \right ) + \ii m \pnabla \bcdot \left ( \bu w^* \right ) + \cc |
| 77 | + = D_0 Z \com |
| 78 | +``` |
| 79 | + |
| 80 | +where $\cc$ denotes the complex conjugate and contraction with $\pnabla = -\p_y \bxh + \p_x \byh$ |
| 81 | +gives the vertical component of the curl. |
| 82 | + |
| 83 | +The baroclinic components obey |
| 84 | + |
| 85 | +```math |
| 86 | +\p_t u - f \v + \p_x p = - \J \left ( \Psi, u \right ) - \bu \bcdot \bnabla U + D_1 u \com \\ |
| 87 | +\p_t \v + f u + \p_y p = - \J \left ( \Psi, \v \right ) - \bu \bcdot \bnabla V + D_1 \v \com \\ |
| 88 | +\p_t p - \tfrac{N^2}{m} w = - \J \left ( \Psi, p \right ) + D_1 p \per |
| 89 | +``` |
| 90 | + |
| 91 | +The dissipation operators are defined |
| 92 | + |
| 93 | +```math |
| 94 | +D_0 = \nu_0 (-1)^{n_0} \nabla^{2n_0} + \mu_0 (-1)^{m_0} \nabla^{2m_0} \com \\ |
| 95 | +D_1 = \nu_1 (-1)^{n_1} \nabla^{2n_1} + \mu_1 (-1)^{m_1} \nabla^{2m_1} |
| 96 | +``` |
| 97 | + |
| 98 | +where $U$ is the barotropic velocity and $u$ is the amplitude of the first baroclinic mode with periodic |
| 99 | +vertical structure $\mathrm{e}^{\mathrm{i} m z}$. |
| 100 | + |
| 101 | +### Implementation |
| 102 | + |
| 103 | +Coming soon. |
| 104 | + |
| 105 | + |
| 106 | +## Vertically Cosine Boussinesq |
| 107 | + |
| 108 | +The vertically-Cosine Boussinesq module solves the Boussinesq system obtained by expanding the |
| 109 | +hydrostatic Boussinesq equations in a Sin/Cos series. The horizontal velocity, for example, becomes |
| 110 | + |
| 111 | +```math |
| 112 | +\uu(x, y, z, t) \mapsto U(x, y, t) + \cos(mz) u(x, y, t) \per |
| 113 | +``` |
| 114 | + |
| 115 | +The horizontal velocity $\vv$, pressure $\pp$, and vertical vorticity $\zzeta$ are also expanded in $\cos(mz)$, |
| 116 | +where $Z = \p_x V - \p_y U$ denotes the barotropic component of the vertical vorticity. The vertical velocity $\ww$ |
| 117 | +and buoyancy $\bb$ are expanded with $\sin(mz)$. |
| 118 | + |
| 119 | +### Basic governing equations |
| 120 | + |
| 121 | +Projecting the vertical vorticity equation onto Sin/Cos modes an equation for the evolution of $Z$, |
| 122 | + |
| 123 | +```math |
| 124 | +\p_t Z + \J \left ( \Psi, Z \right ) |
| 125 | + + \tfrac{1}{2} \bnabla \bcdot \left ( \bu \zeta \right ) + \tfrac{m}{2} \pnabla \bcdot \left ( \bu w \right ) |
| 126 | + = D_0 Z \com |
| 127 | +``` |
| 128 | + |
| 129 | +where $\J(a, b) = (\p_x a)(\p_y b) - (\p_y a)(\p_x b)$ is the Jacobian operator, contraction with $\pnabla = -\p_y \bxh + \p_x \byh$ gives the vertical component of the curl, and $\Psi$ is the barotropic streamfunction defined so that |
| 130 | + |
| 131 | +```math |
| 132 | +\bU = -\p_y\Psi \bxh + \p_x\Psi \byh \qquad \text{and} \qquad Z = \nabla^2 \Psi \per |
| 133 | +``` |
| 134 | + |
| 135 | +The baroclinic components obey |
| 136 | + |
| 137 | +```math |
| 138 | +\p_t u - f \v + \p_x p = - \J \left ( \Psi, u \right ) - \bu \bcdot \bnabla U + D_1u \com \\ |
| 139 | +\p_t \v + f u + \p_y p = - \J \left ( \Psi, \v \right ) - \bu \bcdot \bnabla V + D_1\v \com \\ |
| 140 | +\p_t p - \tfrac{N^2}{m} w = - \J \left ( \Psi, p \right ) + D_1p \per |
| 141 | +``` |
| 142 | + |
| 143 | +The dissipation operators are defined |
| 144 | + |
| 145 | +```math |
| 146 | +D_0 = \nu_0 (-1)^{n_0} \nabla^{2n_0} + \mu_0 (-1)^{m_0} \nabla^{2m_0} \com \\ |
| 147 | +D_1 = \nu_1 (-1)^{n_1} \nabla^{2n_1} + \mu_1 (-1)^{m_1} \nabla^{2m_1} \com |
| 148 | +``` |
| 149 | + |
| 150 | +where $2n_0$ and $2m_0$ are the hyperviscous orders of the arbitrary barotropic dissipation operators |
| 151 | +with coefficients $\nu_0$ and $\mu_0$, while $2n_1$ and $2m_1$ are the orders of the baroclinic |
| 152 | +dissipation operators. |
| 153 | + |
| 154 | +A passive tracer in the Vertically Cosine Boussinesq system is assumed to satisfy a no-flux condition |
| 155 | +at the upper and lower boundaries, and thus expanded in cosine modes so that |
| 156 | + |
| 157 | +```math |
| 158 | +c(x, y, z, t) = C(x, y, t) + \cos(mz) c(x, y, t) \per |
| 159 | +``` |
| 160 | + |
| 161 | +The barotropic and baroclinic passive tracer components then obey |
| 162 | + |
| 163 | +```math |
| 164 | +\p_t C + \J(\Psi, C) + \tfrac{1}{2} \bnabla \bcdot \left ( \bu c \right ) = |
| 165 | + \kappa (-1)^{n_{\kappa}} \nabla^{2{n_{\kappa}}} C \com \\ |
| 166 | +\p_t c + \J(\Psi, c) + \bu \bcdot \bnabla C = \kappa (-1)^{n_{\kappa}} \nabla^{2{n_{\kappa}}} c \com |
| 167 | +``` |
| 168 | + |
| 169 | +where $\kappa$ and $n_{\kappa}$ are the tracer hyperdiffusivity and order of the hyperdiffusivity, respectively. |
| 170 | +The choice $n_{\kappa} = 1$ corresponds to ordinary Fickian diffusivity. |
| 171 | + |
| 172 | +### Implementation |
| 173 | + |
| 174 | +Coming soon. |
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