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6.8: Summary: the equations of motion

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    18073
  • We now have the tools to specify a closed set of equations for most geophysical flows we are likely to encounter. Here we will list the seven equations applicable to seawater:

    \[\frac{D \rho}{D t}=-\rho \vec{\nabla} \cdot \vec{u},\label{eqn:1}\]

    \[\rho \frac{D \vec{u}}{D t}=\rho \vec{g}-\vec{\nabla} p+\mu \nabla^{2} \vec{u}+\mu \vec{\nabla}(\vec{\nabla} \cdot \vec{u}),\label{eqn:2}\]

    \[\rho C_{p} \frac{D T}{D t}=k \nabla^{2} T-\vec{\nabla} \cdot \vec{q}_{r a d}+\rho \varepsilon ; \quad \varepsilon=2 \mathrm{v} e_{i j}^{2},\label{eqn:3}\]

    \[\rho \frac{D S}{D t}=-\vec{\nabla} \cdot \vec{J}_{S} ; \quad \vec{J}_{S}=-\rho \kappa_{S} \vec{\nabla} S,\label{eqn:4}\]

    \[\rho=\rho(p, T, S) \quad \text(the equation of state).\label{eqn:5}\]

    This set involves seven unknowns:

    \[\rho, \vec{u}, p, T, \text { and } S\]

    and the following input parameters:

    \[\vec{g}, \mu, C_{p}, k, q_{r a d}, \text { and } \kappa_{S}.\]

    The second viscosity \(\lambda\) is neglected here as it is usually negligible in geophysical applications.

    In atmospheric flows, the salinity equation is not needed, and the equation of state is the ideal gas law Equation 6.6.5. If moisture effects are important, an advection-diffusion equation for the humidity \(q\) is added (section 6.7), the ideal gas law is modified as in Equation 6.6.9 and the temperature equation Equation \(\ref{eqn:3}\) is extended to include latent heating effects (Curry and Webster 1998).

    We have also neglected the effects of the Earth’s rotation. The flows we deal with here happen on scales of distance and time that we can witness directly, say a few km or less and a few hours or less, and planetary rotation is unimportant for these motions. To deal with larger and/or slower flow phenomena (e.g., synoptic weather systems or ocean currents), we would have to account for the Coriolis and centrifugal accelerations. In the most common approximation, this requires adding a new term to the right-hand side of Equation \(\ref{eqn:2}\): \(f \hat{e}^{(z)}\times\vec{u}\). Here \(\hat{e}^{((z))}\) is the local vertical unit vector and \(f\) = 1.46×10-4s-1 is proportional to the Earth’s rotation rate (e.g., Vallis 2006). We will not go further into these effects in this book, but if you want to experience them directly, try playing catch on a merry-go-round.

    6.8.1 Unpacking the equations of motion

    The equations summarized above contain several vector differential operators of the sort described in section 4.1. Each of these is an abbreviation for one or more partial derivatives acting on one or more components of a vector. It is worthwhile to spend some time looking at more explicit versions of these equations to make sure we understand the operations involved.

    The mass equation Equation 6.2.5 can be written in index notation as

    \[\left(\frac{\partial}{\partial t}+u_{i} \frac{\partial}{\partial x_{i}}\right) \rho=-\rho \frac{\partial u_{i}}{\partial x_{i}}.\]

    Note that \(i\) is a dummy variable to be summed over. Using the familiar notation \(\vec{u}=(u,v,w)\);\(\vec{x}=(x,y,z)\), this can be expanded as

    \[\frac{\partial \rho}{\partial t}+u \frac{\partial \rho}{\partial x}+v \frac{\partial \rho}{\partial y}+w \frac{\partial \rho}{\partial z}=-\rho\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}\right).\]

    Similarly, we can write the momentum equation Equation \(\ref{eqn:2}\) as

    \[\rho\left(\frac{\partial}{\partial t}+u_{i} \frac{\partial}{\partial x_{i}}\right) u_{j}=\rho g_{j}-\frac{\partial p}{\partial x_{j}}+\mu \frac{\partial^{2}}{\partial x_{i}^{2}} u_{j}+\mu \frac{\partial}{\partial x_{j}}\left(\frac{\partial u_{i}}{\partial x_{i}}\right).\]

    Here \(i\) is once again a dummy index, while \(j\) identifies the direction of the velocity component. For the case \(j = 1\) we can write:

    \[\rho\left(\frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}+w \frac{\partial u}{\partial z}\right)=\rho g_{1}-\frac{\partial p}{\partial x}+\mu\left(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}+\frac{\partial^{2} u}{\partial z^{2}}\right)+\mu \frac{\partial}{\partial x}\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}\right).\]

    Exercise: Write out the corresponding equations for the cases \(j = 2\) and \(j = 3\), then repeat the exercise for Equation \(\ref{eqn:3}\) and Equation \(\ref{eqn:4}\). Also, familiarize yourself with the cylindrical and spherical versions summarized in appendix I.