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$\frac{dw}{dt}$ is small compared to $g$, and that the spatial @@ -1444,6 +1444,7 @@ \subsubsection{In the ocean} equation of state for seawater is not easily derived. Instead, we assume that the ocean is a single-component fluid, and we use the density field $\rho$ as the equation of state. + \begin{equation} \rho = \rho(T, S, p) \\ = \rho_0 \left[ 1 - \beta_T(T-T_0) + \beta_S(S-S_0) + \beta_p(p-p_0) \right] @@ -1502,6 +1503,7 @@ \subsubsection{In the ocean} numerical ocean circulation models. \subsection{Nondimensionalization and scaling} +\label{sec:nondimensionalization_and_scaling} A useful technique to simplify the analysis of the governing equations is to scale the variables using characteristic values for each of the variables. @@ -1607,7 +1609,7 @@ \subsection{Exercises} (c) a river inflow into the ocean; (d) a breaking ocean surface wave; (e) water flowing through a pipe with a diameter of 0.1 m and flow speed of 1 m s$^{-1}$. - Assume $nu = 10^{-5} m^2 s^{-1}$ for air and $nu = 10^{-6} m^2 s^{-1}$ for water. + Assume $\nu = 10^{-5} m^2 s^{-1}$ for air and $\nu = 10^{-6} m^2 s^{-1}$ for water. \end{enumerate} @@ -1993,7 +1995,113 @@ \subsection{f-plane and $\beta$-plane approximations} where $f_0 = 2\Omega \sin\theta_0$ and $\beta = \partial f/\partial y = (2\Omega\cos\theta_0) / R_E$ (where $R_E$ is the radius of the Earth). -\subsection{Geostrophic approximation} +\subsection{Geostrophic balance} + +Now that we have incorporated the effects of rotation into our equations of motion, +let's evaluate the scales of the terms in the horizontal momentum equations. +We will start from Eq. \ref{eq:momentum_navier_stokes_rotating}, use the f-plane +notation for the Coriolis term, ignore the viscous terms, and drop the gravity +term as we're looking at the flow in the horizontal plane: + +\begin{equation} + \frac{\partial \mathbf{u}}{\partial t} + + (\mathbf{u} \cdot \nabla) \mathbf{u} + + \mathbf{f} \times \mathbf{u} = + - \frac{1}{\rho} \nabla p +\end{equation} + +As we did in Section \ref{sec:nondimensionalization_and_scaling}, let's scale +each term on the left-hand side with their characteristic scales for mesoscale +ocean flow ($L \sim 10^5\ m$, $T \sim 10^6\ s$, $U \sim 10^{-1}\ m/s$): + +\begin{itemize} + \item $\frac{\partial \mathbf{u}}{\partial t} \sim \frac{U}{T} \sim 10^{-7}$ + \item $(\mathbf{u} \cdot \nabla) \mathbf{u} \sim \frac{U^2}{L} \sim 10^{-7}$ + \item $\mathbf{f} \times \mathbf{u} \sim f_0 U \sim 10^{-6}$ +\end{itemize} +This means that on these oceanic scales ($L \sim 100\ km$, $T \sim 1\ day$), +the inertial terms are of the same order of magnitude as the Coriolis term. +In other words, rotation here is much more important than the local rate of +change or advection. +Also, whatever the scale of the pressure gradient term is, it is the only +term that can balance the rotation. +Thus, if we can state that the inertial terms can be neglected, we can also +state: + +\begin{equation} + \mathbf{f} \times \mathbf{u} \approx - \frac{1}{\rho} \nabla p +\end{equation} +or, in scalar component form: + +\begin{equation} + f u \approx - \frac{1}{\rho} \frac{\partial p}{\partial y} +\end{equation} + +\begin{equation} + f v \approx \frac{1}{\rho} \frac{\partial p}{\partial x} +\end{equation} + +This balance is called the +\textit{geostrophic balance}\index{Geostrophic!balance}\index{Balance!geostrophic}, +and it is a key concept in geophysical fluid dynamics. +It states that the flow is governed by the balance between the rotation and the +pressure gradient force. +Although the geostrophic balance is strictly an approximation and it never holds +exactly, large scale oceanic ($L \sim 100\ km$ and larger) and atmospheric +($L \sim 1000\ km$ and larger) flows are often in geostrophic balance. +For the analysis of geophysical flows at such scales, it is then useful to +define the \textit{geostrophic velocity}\index{Geostrophic!velocity} as: + +\begin{equation} + u_g = - \frac{1}{\rho f} \frac{\partial p}{\partial y} +\end{equation} + +\begin{equation} + v_g = \frac{1}{\rho f} \frac{\partial p}{\partial x} +\end{equation} +Notice that the geostrophic flow is always perpendicular to the pressure gradient, +which means it is parallel to the isobars (lines of constant pressure). +This also means that the isobars are streamlines of the geostrophic flow. +In the northern hemisphere ($f > 0$), the geostrophic flow is cyclonic +(counter-clockwise) around the low-pressure region and anti-cyclonic +(clockwise) around the high-pressure region. +In the southern hemisphere ($f < 0$), it is the opposite. +A nearly geostrophic flow is illustrated in Fig. \ref{fig:geostrophic_flow}. + +\begin{figure}[h] + \centering + \includegraphics[width=0.8\textwidth]{assets/fig_geostrophic_balance.pdf} + \caption{ + Geostrophic flow with a positive value of the Coriolis parameter $f$. + Flow is parallel to the lines of constant pressure (isobars). + Cyclonic flow is anticlockwise around a low pressure region and + anticyclonic flow is clockwise around a high. If $f$ were negative, as in + the Southern Hemisphere, (anti)cyclonic flow would be (anti)clockwise. + This is Fig. 2.5 in AOFD (Vallis, 2017). + } + \label{fig:geostrophic_flow} +\end{figure} + +\subsection{Rossby number} + +Recall that we required the inertial terms to be much smaller than the Coriolis +term for the geostrophic approximation to hold. +Like we did earlier with the Reynolds number to quantify how turbulent a flow is, +we can define the \textit{Rossby number}\index{Rossby!number} as: + +\begin{equation} + \text{Ro} \equiv + \frac{\text{Advection}}{\text{Rotation}} = + \frac{\left( \mathbf{u} \cdot \nabla \right) \mathbf{u}}{\mathbf{f} \times \mathbf{u}} + \approx \frac{\frac{U^2}{L}}{fU} + \approx \frac{U}{fL} +\end{equation} +Though the Rossby number characterizes the relative importance of rotation in +the flow, notice that the rotation term is in the denominator. +The Rossby number is thus small for flows in which rotation dominates over +advection. +In general, flows with a Rossby number of 0.1 or smaller are considered +approximately geostrophically balanced. \subsection{Exercises} @@ -2054,6 +2162,8 @@ \subsection{The Boussinesq equations} \frac{d p_0}{d z} = - \rho_0 g \end{equation} +\subsection{Momentum balance} + Let's first apply the Boussinesq approximation to the momentum balance. Recall the Navier-Stokes equation with rotation (Eq. \ref{eq:momentum_navier_stokes_rotating}), while neglecting the viscosity @@ -2063,10 +2173,10 @@ \subsection{The Boussinesq equations} \frac{d \mathbf{u}}{dt} = - \frac{1}{\rho} \nabla p - f \mathbf{k} \times \mathbf{u} + \mathbf{g} \end{equation} Apply Eqs. \ref{eq:boussinesq_density}-\ref{eq:boussinesq_pressure} to the -above equation: +above equation to get: \begin{equation} - \left( \rho_0 + \delta \rho \right) \left( \frac{d \mathbf{u}}{dt} + f \mathbf{k} \times \mathbf{u} \right) = + \left( \rho_0 + \delta \rho \right) \left( \frac{d \mathbf{u}}{dt} + \mathbf{f} \times \mathbf{u} \right) = - \nabla \left( p_0 + \delta p \right) + \left( \rho_0 + \delta \rho \right) \mathbf{g} \end{equation} @@ -2080,23 +2190,95 @@ \subsection{The Boussinesq equations} on the left-hand side: \begin{equation} - \rho_0 \left( \frac{d \mathbf{u}}{dt} + f \mathbf{k} \times \mathbf{u} \right) = + \rho_0 \left( \frac{d \mathbf{u}}{dt} + \mathbf{f} \times \mathbf{u} \right) = - \nabla \delta p + \delta \rho\ \mathbf{g} \end{equation} \begin{equation} - \frac{d \mathbf{u}}{dt} + f \mathbf{k} \times \mathbf{u} = + \frac{d \mathbf{u}}{dt} + \mathbf{f} \times \mathbf{u} = - \frac{1}{\rho_0} \nabla \delta p + \frac{\delta \rho}{\rho_0} \mathbf{g} \end{equation} -We can express $- g \delta \rho / \rho_0$ as $b$, \textit{buoyancy}\index{buoyancy}: +For convenience of notation, let's now define \textit{buoyancy}\index{buoyancy} +as $b = - g \delta \rho / \rho_0$, and re-write the above to obtain the +Boussinesq momentum equation: \begin{equation} - \frac{d \mathbf{u}}{dt} + f \mathbf{k} \times \mathbf{u} = + \frac{d \mathbf{u}}{dt} + \mathbf{f} \times \mathbf{u} = - \frac{1}{\rho_0} \nabla \delta p + b \mathbf{k} \end{equation} +This equation states that now that we are in a gradually stratified fluid, +the gravity term is scaled by $\delta \rho / \rho_0$ to yield the appropriate +vertical acceleration, and the pressure gradient is due to the relatively +small perturbations in density $\delta \rho$ around the mean density $\rho_0$. + +\subsection{Continuity} + +As we did for the momentum equation, we'll now apply the Boussinesq approximation +(\textit{i.e.} $\rho = \rho_0 + \delta \rho$, $\delta \rho \ll \rho_0$) to the +continuity equation. +Recall the continuity equation in its complete form: + +\begin{equation} + \frac{d \rho}{dt} + \rho \nabla \cdot \mathbf{u} = 0 +\end{equation} +Insert Eq. \ref{eq:boussinesq_density} to get: + +\begin{equation} + \frac{d\delta \rho}{dt} + \left( \rho_0 + \delta \rho \right) \nabla \cdot \mathbf{u} = 0 +\end{equation} +Then, if we can state that that $d\delta \rho / dt \ll \rho_0 \nabla \cdot \mathbf{u}$, +which we will for the Boussinesq approximation, we recover the original +continuity equation for incompressible flows: + +\begin{equation} + \nabla \cdot \mathbf{u} = 0 +\end{equation} +Note that we do not say that strictly $d \delta \rho / dt = 0$, but rather that +we can neglect it in this equation in favor of the velocity divergence term. +The evolution of $\delta \rho$ is still governed by the evolution of buoyancy, +which in turn is governed by the evolution of the temperature and salinity fields +and the equation of state. +The buoyancy $b = - g \delta \rho / \rho_0$ evolves as: + +\begin{equation} + \frac{d b}{dt} = \dot{b} +\end{equation} +and the equation of state can be expressed in terms of buoyancy: + +\begin{equation} + b = b(T, S, p) +\end{equation} +which is just another form of Eq. \ref{eq:equation_of_state_ocean}. + +Finally the temperature and salinity evolve as before, following +Eqs. \ref{eq:temperature_equation_ocean} and \ref{eq:salinity_equation_ocean}, +respectively. + +\subsection{Complete system of equations} + +The full system of Boussinesq equations for the ocean are then: + +\begin{equation} + \frac{d \mathbf{u}}{dt} + \mathbf{f} \times \mathbf{u} = + - \frac{1}{\rho_0} \nabla \delta p + b \mathbf{k} +\end{equation} + +\begin{equation} + \nabla \cdot \mathbf{u} = 0 +\end{equation} + +\begin{equation} + \frac{d T}{dt} = \dot{T} +\end{equation} + +\begin{equation} + \frac{d S}{dt} = \dot{S} +\end{equation} + +\begin{equation} + b = b(T, S, p) +\end{equation} -%\newpage -%\section{Shallow water systems} %\newpage %\section{Boundary layers} @@ -2292,6 +2474,28 @@ \section{Quick reference} \beta = \frac{\partial f}{\partial y} = \frac{2\Omega\cos(\theta_0)}{R_E} \end{equation} +\textbf{Geostrophic balance:} + +\begin{equation} + \mathbf{f} \times \mathbf{u} = - \frac{1}{\rho} \nabla p +\end{equation} + +\textbf{Geostrophic velocity:} + +\begin{equation} + u_g = - \frac{1}{\rho f} \frac{\partial p}{\partial y} +\end{equation} + +\begin{equation} + v_g = \frac{1}{\rho f} \frac{\partial p}{\partial x} +\end{equation} + +\textbf{Rossby number:} + +\begin{equation} + \text{Ro} \equiv \frac{\left( \mathbf{u} \cdot \nabla \right) \mathbf{u}}{\mathbf{f} \times \mathbf{u}} \approx \frac{U}{fL} +\end{equation} + \printindex \end{document}