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2.21.2 Mercator, Nondimensional Equations

The rotating, incompressible, Boussinesq equations of motion Gill [1982] on a sphere can be written in Mercator projection about a latitude $ \theta_0$ and geopotential height $ z=r-r_0$ . The nondimensional form of these equations is:

$\displaystyle {\rm Ro}\frac{D{{\tilde u}}}{Dt} - \frac{{\tilde v} \sin\theta}{\... ...Ro}}{\tilde u}w}{r/H} +\frac{{\rm Ro}{\bf\hat x}\cdot\nabla^2{\bf u}}{{\rm Re}}$ (2.225)

$\displaystyle {\rm Ro}\frac{D{{\tilde v}}}{Dt} + \frac{{\tilde u}\sin\theta}{\s... ...Ro}}{\tilde v}w}{r/H} +\frac{{\rm Ro}{\bf\hat y}\cdot\nabla^2{\bf u}}{{\rm Re}}$ (2.226)


$\displaystyle {\rm Fr}^2\lambda^2\frac{D{w}}{Dt} -b+{\frac{\partial{\pi}}{\partial{z}}} -\frac{\lambda\cot \theta_0 {\tilde u}}{{M_{Ro}}}$ $\displaystyle =$ $\displaystyle \frac{\lambda\mu^2({\tilde u}^2+{\tilde v}^2)}{{M_{Ro}}(r/L)} +\frac{{\rm Fr}^2\lambda^2{\bf\hat z}\cdot\nabla^2{\bf u}}{{\rm Re}}$ (2.227)
$\displaystyle \frac{D{b}}{Dt}+w$ $\displaystyle =$ $\displaystyle \frac{\nabla^2 b}{\Pr{\rm Re}}$  
$\displaystyle \mu^2\left({\frac{\partial{{\tilde u}}}{\partial{x}}} + {\frac{\partial{{\tilde v}}}{\partial{y}}} \right)+{\frac{\partial{w}}{\partial{z}}}$ $\displaystyle =$ 0 (2.228)

Where

$\displaystyle \mu\equiv\frac{\cos\theta_0}{\cos\theta},\ \ \ {\tilde u}=\frac{u^*}{V\mu},\ \ \ {\tilde v}=\frac{v^*}{V\mu}$ (2.229)

$\displaystyle f_0\equiv2\Omega\sin\theta_0,\ \ \ \frac{D}{Dt} \equiv \mu^2\left... ...al y} \right) +\frac{{\rm Fr}^2{M_{Ro}}}{{\rm Ro}} w\frac{\partial}{\partial z}$ (2.230)

$\displaystyle x\equiv \frac{r}{L} \phi \cos \theta_0, \ \ \ y\equiv \frac{r}{L}... ...s \theta_0 {\,\rm d\theta}'}{\cos\theta'}, \ \ \ z\equiv \lambda\frac{r-r_0}{L}$ (2.231)

$\displaystyle t^*=t \frac{L}{V},\ \ \ b^*= b\frac{V f_0{M_{Ro}}}{\lambda}$ (2.232)

$\displaystyle \pi^*=\pi V f_0 L{M_{Ro}},\ \ \ w^*=w V \frac{{\rm Fr}^2\lambda{M_{Ro}}}{{\rm Ro}}$ (2.233)

$\displaystyle {\rm Ro}\equiv\frac{V}{f_0 L},\ \ \ {M_{Ro}}\equiv \max[1,{\rm Ro}]$ (2.234)

$\displaystyle {\rm Fr}\equiv\frac{V}{N \lambda L}, \ \ \ {\rm Re}\equiv\frac{VL}{\nu}, \ \ \ {\rm Pr}\equiv\frac{\nu}{\kappa}$ (2.235)

Dimensional variables are denoted by an asterisk where necessary. If we filter over a grid scale typical for ocean models ( $ 1m<L<100km$ , $ 0.0001<\lambda<1$ , $ 0.001m/s <V<1 m/s$ , $ f_0<0.0001 s^{-1}$ , $ 0.01 s^{-1}<N<0.0001 s^{-1}$ ), these equations are very well approximated by
$\displaystyle {\rm Ro}{\frac{D{{\tilde u}}}{Dt} }- \frac{{\tilde v} \sin\theta}{\sin\theta_0}+{M_{Ro}}{\frac{\partial{\pi}}{\partial{x}}}$ $\displaystyle =$ $\displaystyle -\frac{\lambda{\rm Fr}^2{M_{Ro}}\cos \theta}{\mu\sin\theta_0} w +\frac{{\rm Ro}\nabla^2{{\tilde u}}}{{\rm Re}}$ (2.236)
$\displaystyle {\rm Ro}\frac{D{{\tilde v}}}{Dt} + \frac{{\tilde u}\sin\theta}{\sin\theta_0}+{M_{Ro}}{\frac{\partial{\pi}}{\partial{y}}}$ $\displaystyle =$ $\displaystyle \frac{{\rm Ro}\nabla^2{{\tilde v}}}{{\rm Re}}$ (2.237)
$\displaystyle {\rm Fr}^2\lambda^2\frac{D{w}}{Dt} -b+{\frac{\partial{\pi}}{\partial{z}}}$ $\displaystyle =$ $\displaystyle \frac{\lambda\cot \theta_0 {\tilde u}}{{M_{Ro}}} +\frac{{\rm Fr}^2\lambda^2\nabla^2w}{{\rm Re}}$ (2.238)
$\displaystyle \frac{D{b}}{Dt}+w$ $\displaystyle =$ $\displaystyle \frac{\nabla^2 b}{\Pr{\rm Re}}$ (2.239)
$\displaystyle \mu^2\left({\frac{\partial{{\tilde u}}}{\partial{x}}} + {\frac{\partial{{\tilde v}}}{\partial{y}}} \right)+{\frac{\partial{w}}{\partial{z}}}$ $\displaystyle =$ 0 (2.240)
$\displaystyle \nabla^2$ $\displaystyle \approx$ $\displaystyle \left(\frac{\partial^2}{\partial x^2} +\frac{\partial^2}{\partial y^2} +\frac{\partial^2}{\lambda^2\partial z^2}\right)$ (2.241)

Neglecting the non-frictional terms on the right-hand side is usually called the 'traditional' approximation. It is appropriate, with either large aspect ratio or far from the tropics. This approximation is used here, as it does not affect the form of the eddy stresses which is the main topic. The frictional terms are preserved in this approximate form for later comparison with eddy stresses.

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Next: 3. Getting Started with Up: 2.21 Nonlinear Viscosities for Previous: 2.21.1 Eddy Viscosity   Contents
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