2. Conservation of momentum (i)#

Questions to be answered

  1. What is the equation for conservation of momentum in the atmospheric boundary layer?

  2. Which assumptions are made to get to this equation?

  3. What is the meaning of the Reynolds and the Rossby number?

  4. What are the values of the Reynolds and the Rossby number and what is the implication thereof?

\(\def\pd#1#2{\dfrac{\partial #1}{\partial #2}}\) \(\def\rhobar{\overline{\rho}}\) \(\def\pdd#1#2{\dfrac{\partial^2 #1}{\partial#2^2}}\) \(\def\uibar{\overline{u}_i}\) \(\def\ujbar{\overline{u}_j}\) \(\def\uiprime{u_i^\prime}\) \(\def\ujprime{u_j^\prime}\) \(\def\ubar{\overline{u}}\) \(\def\vbar{\overline{v}}\) \(\def\wbar{\overline{w}}\)

2.1. Conservation of momentum in an incompressible atmospheric boundary layer.#

Wind is a key ingredient of boundary layer meteorology. The evolution of wind speed is generally covered via the conservation of momentum, also known as the Navier-Stokes equations.

First, we introduce the equations for horizontal momentum for an incompressible flow, where we have already introduced the mean density \(\rhobar\), and we will also split the other thermodynamic variable \(p\) in a mean and a perturbation.

\[\begin{split} \pd{u}{t} + u_j \pd{u}{x_j} &= - \dfrac{1}{\rhobar} \pd{\overline{p}}{x} - \dfrac{1}{\rhobar} \pd{p^\prime}{x} + f v + \nu \pdd{u}{x_j} \\ \pd{v}{t} + u_j \pd{v}{x_j} &= - \dfrac{1}{\rhobar} \pd{\overline{p}}{y} - \dfrac{1}{\rhobar} \pd{p^\prime}{y} - f u + \nu \pdd{v}{x_j} \end{split}\]

We can now substitute the mean pressure gradient by the geostrophic wind definitions \(fv_g \equiv \dfrac{1}{\rhobar}\pd{p}{x}\) and \(fu_g \equiv - \dfrac{1}{\rhobar}\pd{p}{y}\) to get:

\[\begin{split} \pd{u}{t} + u_j \pd{u}{x_j} &= - \dfrac{1}{\rhobar} \pd{p^\prime}{x} + f \left( v - v_g \right) + \nu \pdd{u}{x_j} \\ \pd{v}{t} + u_j \pd{v}{x_j} &= - \dfrac{1}{\rhobar} \pd{p^\prime}{y} - f \left( u - u_g \right) + \nu \pdd{v}{x_j} \end{split}\]

For the vertical momentum equations, we briefly take a step back and reintroduce the density perturbation and multiply the equation with \(\rho\):

\[ \left( \rhobar + \rho^\prime \right) \pd{w}{t} + \left( \rhobar + \rho^\prime \right) u_j \pd{w}{x_j} = - \pd{\overline{p}}{z} - \pd{p^\prime}{z} - \left( \rhobar + \rho^\prime \right) g + \nu \pdd{w}{x_j} \]

We know that in on the larger scales, the flow is in hydrostatic balance \(\pd{\overline{p}}{z} = - \rhobar g\), and hence we can eliminate these very large terms from the equation. If we divide out \(\rho\) again, and neglect perturbations, we get:

\[ \pd{w}{t} + u_j \pd{w}{x_j} = - \dfrac{1}{\rhobar} \pd{p^\prime}{z} - \dfrac{\rho^\prime}{\rhobar} g + \nu \pdd{w}{x_j} \]

With index notation, we can combine the three momentum equations:

\[ \pd{u_i}{t} + u_j \pd{u_i}{x_j} = - \dfrac{1}{\rhobar} \pd{p^\prime}{x_i} + f \epsilon_{ij3} \left( u_j - u_{g,j} \right) - \delta_{i3} \dfrac{\rho^\prime}{\rhobar} g + \nu \pdd{u_i}{x_j} \]

The last step is to put the advection term in the flux form. With the help of the chain rule, we can write:

\[ \pd{u_i u_j}{x_j} = u_j \pd{u_i}{x_j} + u_i \pd{u_j}{x_j} \]

The latter term is zero because of incompressibility, thus we can write our final equation as:

\[ \pd{u_i}{t} + \pd{u_i u_j}{x_j} = - \dfrac{1}{\rhobar} \pd{p^\prime}{x_i} + f \epsilon_{ij3} \left( u_j - u_{g,j} \right) - \delta_{i3} \dfrac{\rho^\prime}{\rhobar} g + \nu \pdd{u_i}{x_j} \]

2.2. Reynolds averaging the velocity components#

We will now introduce the Reynolds averaging of the velocity components, thus we split all velocity instances in a mean and a perturbation.

\[\begin{split} \pd{\uibar}{t} + \pd{\uiprime}{t} + \pd{\uibar\ujbar}{x_j} + \pd{\uiprime \ujbar}{x_j} + \pd{\uibar \ujprime}{x_j} + \pd{\uiprime \ujprime}{x_j} = \\ - \dfrac{1}{\rhobar} \pd{p^\prime}{x_i} + f \epsilon_{ij3} \left( \ujbar + \ujprime - u_{g,j} \right) - \delta_{i3} \dfrac{\rho^\prime}{\rhobar} g + \nu \pdd{\uibar}{x_j} + \nu \pdd{\uiprime}{x_j} \end{split}\]

We are interested to retrieve the evolution of the mean velocity \(\uibar\), so we take the mean of the entire equation.

\[\begin{split} \overline{ \pd{\uibar}{t} + \pd{\uiprime}{t} + \pd{\uibar \ujbar}{x_j} + \pd{\uiprime \ujbar}{x_j} + \pd{\uibar \ujprime}{x_j} + \pd{\uiprime \ujprime}{x_j} = } \\ \overline {- \dfrac{1}{\rhobar} \pd{p^\prime}{x_i} + f \epsilon_{ij3} \left( \ujbar + \ujprime - u_{g,j} \right) - \delta_{i3} \dfrac{\rho^\prime}{\rhobar} g + \nu \pdd{\uibar}{x_j} + \nu \pdd{\uiprime}{x_j}} \end{split}\]

Subsequently, we make use of the rule \(\overline{A + B} = \overline{A} + \overline{B}\).

\[\begin{split} \overline{ \pd{\uibar}{t} } + \overline{ \pd{\uiprime}{t} } + \overline{ \pd{\uibar \ujbar}{x_j} } + \overline{ \pd{\uiprime \ujbar}{x_j} } + \overline{ \pd{\uibar \ujprime}{x_j} } + \overline{ \pd{\uiprime \ujprime}{x_j} } = \\ \overline{ - \dfrac{1}{\rhobar} \pd{p^\prime}{x_i} } + \overline{ f \epsilon_{ij3} \left( \ujbar + \ujprime - u_{g,j} \right) } - \overline{ \delta_{i3} \dfrac{\rho^\prime}{\rhobar} g } + \overline{ \nu \pdd{\uibar}{x_j} } + \overline{ \nu \pdd{\uiprime}{x_j} } \end{split}\]

Subsequently, we make use of the rules \(\overline{\pd{A}{t}} = \pd{\overline{A}}{t}\), \(\overline{A^\prime} = 0\), \(\overline{\overline{A} + B} = \overline{A} + \overline{B}\), and \(\overline{cA} = c \overline{A}\) to end at

\[ \pd{\uibar}{t} + \pd{\uibar \ujbar}{x_j} = - \pd{\overline{ \uiprime \ujprime }}{x_j} + f \epsilon_{ij3} \left( \ujbar - u_{g,j} \right) + \nu \pdd{\uibar}{x_j} \]

If we define velocity scale \(U\), length scale \(L\), time scale \(L / U\), then we can estimate a typical magnitude of the left-hand side terms and the turbulent flux term as \(U^2/L\), \(f U\) for the velocity term, and \(\nu U / L^2\) for the viscous term. We can compute now the dimensionless ratio that compares the magnitude of the advection to the viscous forces as

\[ \dfrac{U^2}{L} \dfrac{L^2}{\nu U} = \dfrac{U L}{\nu} = Re \]

and do the same for the ratio of advection to rotation:

\[ \dfrac{U^2}{L} \dfrac{1}{f U} = \dfrac{U}{f L} = Ro \]

In that way, we have defined the Reynolds number \(Re\) and Rossby number \(Ro\). Taking \(U = 1\), \(L = 10^3\), and \(f = 10^{-4}\), we find that \(Re = 10^8\), and \(Ro = 10\). Therefore, the viscous force can be safely neglected, but the Coriolis force cannot.

The final conservation of momentum equation is:

\[ \pd{\uibar}{t} + \pd{\uibar \ujbar}{x_j} = - \pd{\overline{ \uiprime \ujprime }}{x_j} + f \epsilon_{ij3} \left( \ujbar - u_{g,j} \right) \]

In order to study turbulence, we often simplify this set even further, by assuming horizontal homogeneity (except for the large-scale pressure force) and assuming that \(\overline{w} = 0\), thus no subsidence. The set then reduces to

\[\begin{split} \pd{\ubar}{t} &= - \pd{\overline{ u^\prime w^\prime }}{z} + f \left( \vbar - v_g \right) \\ \pd{\vbar}{t} &= - \pd{\overline{ v^\prime w^\prime }}{z} - f \left( \ubar - u_g \right) \end{split}\]

This set of equation has been the starting point of the important work of Ekman.