$$ \def\ab{\boldsymbol{a}} \def\bb{\boldsymbol{b}} \def\eb{\boldsymbol{e}} \def\fb{\boldsymbol{f}} \def\gb{\boldsymbol{g}} \def\kb{\boldsymbol{k}} \def\nb{\boldsymbol{n}} \def\ub{\boldsymbol{u}} \def\vb{\boldsymbol{v}} \def\xb{\boldsymbol{x}} \def\Xb{\boldsymbol{X}} \def\Ab{\boldsymbol{A}} \def\Bb{\boldsymbol{B}} \newcommand{\Cb}{\boldsymbol{C}} \newcommand{\Db}{\boldsymbol{D}} \def\Eb{\boldsymbol{E}} \def\Fb{\boldsymbol{F}} \def\Jb{\boldsymbol{J}} \def\Ub{\boldsymbol{U}} \def\xib{\boldsymbol{\xi}} \def\evx{\boldsymbol{e}_x} \def\evy{\boldsymbol{e}_y} \def\evz{\boldsymbol{e}_z} \def\evr{\boldsymbol{e}_r} \def\evt{\boldsymbol{e}_\theta} \def\evp{\boldsymbol{e}_r} \def\evf{\boldsymbol{e}_\phi} \def\evb{\boldsymbol{e}_\parallel} \def\omb{\boldsymbol{\omega}} \def\dA{\;\mathrm{d}\Ab} \def\dS{\;\mathrm{d}\boldsymbol{S}} \def\dV{\;\mathrm{d}V} \def\dl{\mathrm{d}\boldsymbol{l}} \def\bfzero{\boldsymbol{0}} \def\Rey{\mathrm{Re}} \newcommand{\dds}[1]{\frac{\mathrm{d}{#1}}{\mathrm{d}s}} \newcommand{\ddy}[2]{\frac{\partial{#1}}{\partial{#2}}} \newcommand{\ddt}[1]{\frac{\mathrm{d}{#1}}{\mathrm{d}t}} \newcommand{\DDt}[1]{\frac{\mathrm{D}{#1}}{\mathrm{D}t}} \newcommand{\gradH}{\nabla_\mathrm{H}} \newcommand{\grad}{\nabla} \newcommand{\Dl}{\mathrm{\Delta}} \newcommand{\Omb}{\boldsymbol{\mathrm{\Omega}}} \newcommand{\Omrm}{\mathrm{\Omega}} \newcommand{\Ld}{\mathcal{L}_d } \newcommand{\nd}[1]{{#1}^*} \newcommand{\vh}{\nd{\boldsymbol{v}}} \newcommand{\bh}{\nd{b}} \newcommand{\ph}{\nd{p}} \newcommand{\fh}{\nd{\fb}} \newcommand{\tha}{\nd{t}} \newcommand{\wh}{\nd{w}} \newcommand{\zh}{\nd{z}} \newcommand{\gradHh}{\nd{\nabla}_H} \newcommand{\psih}{\nd{\psi}} \newcommand{\Ro}{\mathrm{Ro}} \newcommand{\ddz}[1]{\frac{\mathrm{d}{#1}}{\mathrm{d}z}} $$
2 Stratification
2.1 Buoyancy frequency
A dense object sinks in a fluid because of gravity, whereas an object that is less dense than the surrounding fluid tends to rise because of an upward force exerted by the fluid. This force – known as buoyancy – is associated with the weight of the displaced fluid.
Typically, fluids in the atmosphere and ocean are stratified; that is, their density varies with height. In a stably stratified medium, where lower-density fluid lies above higher-density fluid, a fluid parcel that is displaced vertically tends to return towards its original position because of buoyancy. By considering the motion of such a fluid parcel, we can define the buoyancy frequency, an important property of stratified fluids.
Consider a fluid at rest with density \(\tilde\rho(z)\) and hydrostatic pressure \(\tilde p\), i.e., \[ \frac{d\tilde p}{dz}=-g\tilde\rho. \] Now consider raising a parcel of fluid from \(z=0\) to \(z=Z\) (see Figure 2.1). The pressure of the parcel adjusts to the pressure of the surrounding environment, but its density may differ from that of the environment. To make this distinction, we denote the density of the parcel by \(\rho_p\). At \(z=0\), the parcel and the environment have the same density, and hence \[ \frac{d\tilde p}{dz}\bigg|_{z=0} =-g\tilde\rho(0) =-g\rho_p(0). \]
The environment at \(z=Z\) is also in hydrostatic balance: \[ \frac{d\tilde p}{dz}\bigg|_{z=Z} =-g\tilde\rho(Z). \] However, the displaced parcel is no longer in hydrostatic balance. Since we consider a fixed mass of incompressible fluid, the parcel density does not change when it is lifted: \[ \rho_p(Z)=\rho_p(0)=\tilde\rho(0). \tag{2.1}\] The upward pressure-gradient force per unit volume acting on the parcel is \[ -\frac{d\tilde p}{dz}\bigg|_{z=Z}=g\tilde\rho(Z), \] whereas its weight per unit volume is \(g\rho_p(Z)\). Hence, the net vertical force per unit volume is \[ F_{\mathrm{b}} =-\frac{d\tilde p}{dz}\bigg|_{z=Z}-g\rho_p(Z) =g\left[\tilde\rho(Z)-\rho_p(Z)\right]. \tag{2.2}\] We can view \(F_{\mathrm{b}}\) as the net buoyancy force arising from the density difference between the parcel and its environment.
We can then apply Newton’s second law to the parcel: \[ \rho_p(Z)\frac{d^2 Z}{dt^2}=F_{\mathrm{b}}, \] and hence \[ \frac{d^2 Z}{dt^2} = \frac{g\left[\tilde\rho(Z)-\rho_p(Z)\right]} {\rho_p(Z)}. \tag{2.3}\] Using Equation 2.1, we can write Equation 2.3 as \[ \frac{d^2 Z}{dt^2} = \frac{g\left[\tilde\rho(Z)-\tilde\rho(0)\right]} {\tilde\rho(0)} \approx \frac{g}{\tilde\rho(0)} \frac{d\tilde\rho}{dz}\bigg|_{z=0}Z, \tag{2.4}\] where we have used a Taylor expansion of \(\tilde\rho(Z)\) about \(z=0\).
Defining \[ N_0^2 = -\frac{g}{\tilde\rho(0)} \frac{d\tilde\rho}{dz}\bigg|_{z=0}, \] the equation becomes \[ \frac{d^2 Z}{dt^2}=-N_0^2Z. \tag{2.5}\]
We will see in the example below that if \({d\tilde\rho}/{dz}<0\), the solution of the above equation is oscillatory and \(N_0\) is the frequency of this motion.
We could place our parcel at any height, rather than at \(z=0\), and therefore generalise Equation 2.5 to define the general buoyancy frequency \[ N^2(z) = -\frac{g}{\tilde\rho}\frac{d\tilde\rho}{dz}. \tag{2.7}\] The quantity \(N(z)\) is known as the buoyancy frequency or Brunt–Väisälä frequency. It gives the frequency of small vertical oscillations of a fluid parcel in a stably stratified medium.
If \({d\tilde\rho}/{dz}>0\) then \(N^2<0\) and the stratification is unstable: a parcel displaced upwards continues to rise rather than oscillating about its original position.
2.2 Boussinesq equations
2.2.1 Density and temperature variation in the ocean and atmosphere
In the lower atmosphere and most parts of the ocean, the density (or temperature) does not change significantly at different horizontal locations \(\xb=(x,y)\) or in time. This motivates the Boussinesq approximation which involves writing the density (and pressure) as a sum of a base profile plus small perturbation, where the base profile only depends on \(z\). We will formulate this approximation mathematically in the next section, but before that it will be insightful to look at few observations of density and temperature variations in the ocean and atmosphere.
Figure 2.2 shows a combination of two types of measurements in a region in the arctic ocean. In panel (a), there are four red dots that are mooring measurements obtained by a collection of devices connected to a wire and anchored on the sea floor. The other dots are measurements taken by a cruise that sends diving devices down into the ocean. The details of these observations are not important for us, but they show that several locations are sampled by different measuring devices and at different times. Panel (b) shows the density profiles of the cruise measurements at different locations. It is clear that the density profile has a (horizontal) mean that is a function of \(z\). The perturbations to this mean by changing the horizontal location are much smaller than the mean value itself. The variation of density from the base profile is better quantified in the panel (c). The solid black curve shows the mean (base) profile of the density next to a single cast with dashed black curve, which marginally deviates from the mean. Panel (d) shows the density profiles for a fixed location (location D in panel (a)), but at different times. It can be inferred from this plot that changing time does not affect the mean density profile very much. Looking at different part of the ocean, we get a more general base profile (a smoother mean). Nevertheless, the underlying message of this example still holds: the density is well approximated with a base profile that is a function of depth only plus small perturbations.
Figure 2.3 displays the (daily) temperature profiles over a region in India. Atmospheric temperature is directly related to the density at each altitude (which has a fixed pressure). Hence, the relative magnitude of the mean temperature to perturbations can translate to the density as well. It is evident from this figure that changes in temperature at each altitude are relatively small for lower atmosphere (below 15km). There is a jump in temperature that is known as “tropopause”. Near the tropopause, the deviations from the mean temperature profile are larger, and the Boussinesq approximation is not justified.
Measurements similar to the examples just discussed are abundant in the literature thanks to several decades of observational campaigns. Considering them all together, we can conclude that in most parts of the ocean, and in the lower atmosphere, the variations in density at a fixed \(z\) are small relative to their means.
2.2.2 Boussinesq approximation
As we have seen, the density in the ocean and in the lower layer of atmosphere varies by only a few percent across the depth or altitude. We can therefore exploit the smallness of these variations and write \[ \rho(\xb,t) = \rho_0 + \delta\rho(\xb,t) \] where \(\rho_0\) is the average density (a constant) and \(\delta\rho\) are (small) perturbations away from this mean. The perturbations are assumed small so that \(|\delta\rho|\ll\rho_0\). Substituting this into the inviscid, incompressible momentum equation (i.e., Equation 1.7 with \(\nu=0\)) and again assuming gravity is the only body force acting so that \(\fb_b=-g\eb_z\), we find \[ (\rho_0 + \delta\rho(\xb,t)) \frac{D\ub}{Dt} = -\nabla p - (\rho_0 + \delta\rho(\xb,t))g\eb_z. \tag{2.8}\] Separating the horizontal and vertical momentum equations in Equation 2.8 and dividing them by \(\rho_0\) yields \[ \Big(1 + \underbrace{\frac{\delta\rho}{\rho_0}}_{\mathcal{O}(\epsilon)}\Big) \frac{D\vb}{Dt} = - \frac{1}{\rho_0} \gradH p, \tag{2.9}\] \[ \Big(1 + \underbrace{\frac{\delta\rho}{\rho_0}}_{\mathcal{O}(\epsilon)}\Big) \underbrace{\frac{D w}{Dt}}_{\mathcal{O}(\epsilon)} = - \frac{1}{\rho_0} \frac{\partial p}{\partial z} - \Big(1 + \underbrace{\frac{\delta\rho}{\rho_0}}_{\mathcal{O}(\epsilon)}\Big) g, \tag{2.10}\] where \(\vb = (u,v)\) and \(\gradH = (\partial/\partial x,\partial/\partial y)\) are the horizontal velocity and gradient, respectively.
In most geophysical flows, both \(\delta\rho/\rho_0\) and the vertical acceleration \(Dw/Dt\) are relatively small terms and we denote them by \(\mathcal{O}(\epsilon)\) for \(0<\epsilon \ll 1\), while we assume all other terms to be \(\mathcal{O}(1)\). We can ignore the \(\mathcal{O}(\epsilon)\) terms in Equation 2.9 to obtain \[ \frac{D\vb}{Dt} = - \frac{1}{\rho_0} \gradH p. \tag{2.11}\] Following a similar approach for Equation 2.10 and keeping only the \(\mathcal{O}(1)\) terms (i.e. ignoring the \(\mathcal{O}(\epsilon)\) and \(\mathcal{O}(\epsilon^2)\) terms) leads to \[ \frac{1}{\rho_0} \frac{\partial p}{\partial z} = - g. \] This is simply the leading-order hydrostatic balance, which is too simplistic for the applications we are considering. To retain the vertical dynamics and the effects of density perturbations, we must keep the \(\mathcal{O}(\epsilon)\) terms, while ignoring the \(\mathcal{O}(\epsilon^2)\) terms: \[ \frac{D w}{Dt} = - \frac{1}{\rho_0} \frac{\partial p}{\partial z} - \Big(1 + \frac{\delta\rho}{\rho_0}\Big) g = - \frac{1}{\rho_0} \frac{\partial p}{\partial z} - \frac{\rho}{\rho_0} g. \tag{2.12}\] The approximation that leads to Equation 2.11 and Equation 2.12 is known as the Boussinesq approximation. Combining these momentum equations with (1.4) and (1.6), we can write \[ \frac{D\ub}{Dt} = -\frac{1}{\rho_0}\nabla p -\frac{\rho}{\rho_0}g\eb_z. \tag{2.13}\] \[ \nabla\cdot\ub = 0, \tag{2.14}\] \[ \frac{D\rho}{Dt} = \frac{\partial \rho}{\partial t} + (\ub\cdot\nabla)\rho= 0, \tag{2.15}\] which are known as the Boussinesq equations written in terms of the density variable.
Although we have derived this set of Boussinesq equations starting from the assumption of an incompressible fluid, it is possible to show that the Boussinesq approximation applies to a more general compressible fluid.
The Boussinesq approximation is valid almost everywhere in the ocean since density variations are small. However, it is also valid for any motions in the atmosphere that are confined to a layer that is thin compared with the scale height of the atmosphere. For the Earth’s atmosphere, we have seen that the scale height is approximately \(8\)km and so the Boussinesq approximation should apply to motions that are confined to layers of about \(1\)km or less.
2.2.3 Buoyancy as a variable
Next, let’s consider splitting our variables into a reference state and perturbations to this state. We will consider a static, time-independent reference state with a vertical density stratification \(\rho=\tilde\rho(z)\). Equation 2.14 and Equation 2.15 are automatically satisfied by this reference state and Equation 2.13 demands \[ p=\tilde p(z) \quad \text{with} \quad \frac{d\tilde p}{dz} = -\tilde\rho g. \tag{2.16}\] Decomposing our variables in this way gives \[ \ub = \ub'(\xb,t), \quad \rho = \tilde\rho(z) + \rho'(\xb,t), \quad p = \tilde p(z) + p'(\xb,t) \tag{2.17}\] where a tilde denotes a reference state variable and the primed quantities are perturbations to these states (for now no assumptions are made about the size of the perturbations).
Now, substituting (2.17) into Equation 2.13 gives \[ \frac{\partial\ub'}{\partial t} + (\ub'\cdot\nabla)\ub' = -\frac{1}{\rho_0}\nabla p' -\frac{\rho'}{\rho_0}g\eb_z = -\frac{1}{\rho_0}\nabla p' +b\eb_z, \] where we have used Equation 2.16 to cancel reference state terms and introduced the buoyancy variable \[ b = -\frac{\rho'}{\rho_0} g . \tag{2.18}\] Note that this expression is analogous to the buoyancy force per unit mass \(F_b/\rho_0\) introduced in Equation 2.2.
Substituting (2.17) into Equation 2.14 simply gives \(\nabla\cdot\ub'=0\). Finally, substituting (2.17) into Equation 2.15 gives \[ \frac{\partial(\tilde \rho(z)+\rho')}{\partial t} + (\ub'\cdot\nabla)(\tilde\rho(z)+\rho')=0\] \[ \Rightarrow \frac{\partial\rho'}{\partial t} + (\ub'\cdot\nabla)\rho' + w'\frac{d\tilde\rho}{dz}=0. \tag{2.19}\] Multiplying Equation 2.19 by \(-\frac{g}{\rho_0}\) allows us to form \[\begin{align*} \frac{Db}{Dt}+\left(\frac{-g}{\rho_0}\frac{d\tilde\rho}{dz}\right)w'&=0,\\ \Rightarrow\frac{Db}{Dt}+N^2(z)w'&=0 \end{align*}\] where \[ N^2(z) = -\frac{g}{\rho_0}\frac{d\tilde\rho}{dz}. \] Here \(N(z)\) is the buoyancy frequency in the Boussinesq approximation. Note it is subtly different to the definition in Equation 2.7.
Gathering together the Boussinesq equations for the perturbation quantities we have the following equation set:
where we have dropped the primes from perturbation quantities as is customary.
Note here that \(p\) is a perturbation to the background pressure state. In general, when \(p\) appears along side buoyancy in the governing equations then this indicates it is a perturbation pressure. This is in contrast to the Boussinesq equations where \(p\) in Equation 2.13 is taken to mean the full pressure (likewise in Equation 2.13, \(\rho\) is the full density).
The buoyancy (Brunt–Väisälä) frequency that appears in Equation 2.22 is an important quantity in different regimes of geophysical flows. Figure 2.4 shows the values of \(N^2\) for a region in the atmosphere/ocean as functions of altitude/depth. In the left plot, \(N^2\) over the region of Gadanki in India is shown. There is a noticeable jump at around 18km altitude (similar to what we observed in figure Figure 2.3, marking the “tropopause”, which is the border between two layers of atmosphere (troposphere and stratosphere). Note that the height of tropopause varies to some degrees in different locations (that is why the jump in Figure 2.3 and Figure 2.4 appears at slightly different altitudes). Around the tropopause, constant \(N\) or even the Boussinesq approximation are not valid, but in the lower atmosphere (below the tropopause) \(N\) can be assumed constant. Horizontal bars show standard deviations obtained while considering 43 different measurements during April 2006 to March 2007. The relatively small variability in the lower atmosphere supports representing the stratification by a slowly varying background profile. Near the tropopause, however, \(N\) varies rapidly with height and a constant-\(N\) approximation is no longer appropriate.
The right plot is obtained from a mooring observation (“NISKINe” campaign). For the deep ocean (lower than 800 meters), a constant-\(N\) approximation appears reasonable. However, in the upper ocean the variation of \(N\) with depth cannot be neglected. A comparison between the two figures shows that \(N\) is an order of magnitude smaller in the ocean.
2.2.4 Mountain waves caused by stratification
One of the interesting phenomena that arises as a result of stratification is the generation of mountain waves (also known as lee waves). We study a simplified version of these waves using the Boussinesq equations derived in the previous section. Consider a gentle mountain (see Figure 2.5), by which we mean that the topographic variations, or equivalently the height of the mountain, are relatively small. We assume that the mountain is periodic and described by \[ h = h_0 \sin(kx), \tag{2.23}\] where \(h_0\) is the height of the mountain. A wind of speed \(U\), which we assume to be constant, blows over the mountain and displaces air parcels vertically. We therefore expect buoyancy to act as a restoring force on these displaced parcels. We also assume that the buoyancy frequency \(N\) is constant.
Let us consider a steady, two-dimensional version of this problem, assuming that variations in the \(y\) direction are negligible. We write the flow variables as the sum of a background wind and disturbances caused by the mountain (denoted by primes): \[ u = U + u', \qquad v=0, \qquad w = w', \qquad p = p', \qquad b = b'. \] Substituting these expressions into (2.20)-(2.22), and neglecting \(\partial/\partial y\) and \(\partial/\partial t\), gives \[ \begin{aligned} U \frac{\partial u'}{\partial x} + u' \frac{\partial u'}{\partial x} + w' \frac{\partial u'}{\partial z} &= -\frac{1}{\rho_0}\frac{\partial p'}{\partial x}, \\ U \frac{\partial w'}{\partial x} + u' \frac{\partial w'}{\partial x} + w' \frac{\partial w'}{\partial z} &= -\frac{1}{\rho_0}\frac{\partial p'}{\partial z} + b',\\ \frac{\partial u'}{\partial x} + \frac{\partial w'}{\partial z} &= 0, \\ U \frac{\partial b'}{\partial x} + N^2 w' &= 0. \end{aligned} \]
If the mountain is small, it generates small disturbances. We can therefore neglect products of disturbance quantities, leading to the linearised equations. Dropping the primes for convenience, we obtain \[ \begin{aligned} U \frac{\partial u}{\partial x} &= -\frac{1}{\rho_0}\frac{\partial p}{\partial x}, \\ U \frac{\partial w}{\partial x} &= -\frac{1}{\rho_0}\frac{\partial p}{\partial z} + b, \\ \frac{\partial u}{\partial x} + \frac{\partial w}{\partial z} &= 0, \\ U \frac{\partial b}{\partial x} + N^2 w &= 0. \end{aligned} \tag{2.24}\]
Eliminating \(u\), \(p\) and \(b\) in favour of \(w\) in Equation 2.24, we obtain \[ \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial z^2} = -\frac{N^2}{U^2}w. \tag{2.25}\]
At the ground, a fluid parcel must follow the terrain. Since the surface is described by \(z=h(x)\), the vertical velocity of a parcel on the surface is \[ w = \frac{Dh}{Dt}. \] For a steady, two-dimensional flow this gives \[ w(x,z=h) = (U+u')\frac{\partial h}{\partial x}. \] For a small mountain, we linearise this condition by evaluating it at \(z=0\) and neglecting the product \(u'\partial h/\partial x\). Hence, \[ w(x,z=0) \approx U\frac{\partial h}{\partial x}. \tag{2.26}\] Using Equation 2.23, the lower boundary condition becomes \[ w(x,z=0) = h_0Uk\cos(kx). \tag{2.27}\]
Motivated by Equation 2.27, we seek a solution of the form \[ w(x,z) = h_0Uk\cos(kx+mz). \] Substitution into Equation 2.25 gives \[ \left(-k^2-m^2+\frac{N^2}{U^2}\right) h_0Uk\cos(kx+mz)=0. \] For this expression to hold for all \(x\) and \(z\), we require \[ -k^2-m^2+\frac{N^2}{U^2}=0. \] Hence, \[ m^2 = \frac{N^2}{U^2}-k^2, \qquad m = \pm\sqrt{\frac{N^2}{U^2}-k^2}. \tag{2.28}\] Here we have assumed that \(N/U>k\), so that \(m\) is real and the disturbance can propagate vertically as a mountain wave. This condition is favoured by strong stratification, weak winds, or sufficiently long horizontal wavelengths.
We now need to determine which sign of \(m\) corresponds to the physical solution. Since the waves are generated by the mountain, we expect their energy to propagate upwards, away from the ground. The vertical flux of wave energy is given by the pressure work \(pw\).
For our solution \(w=h_0Uk\cos(kx+mz)\), the continuity equation gives \[ u=-h_0Um\cos(kx+mz)+C(z), \] where \(C(z)\) is independent of \(x\). The term \(C(z)\) represents a horizontally uniform (\(k=0\)) component of the flow. Since the mountain forcing contains only the wavenumber \(k\), and the linearised equations do not couple different horizontal wavenumbers, this component is not generated by the mountain. Moreover, the horizontally uniform flow is already included in our prescribed background wind. We therefore set \(C(z)=0\), giving \[ u=-h_0Um\cos(kx+mz). \] The \(x\)-momentum equation then gives \[ p=-\rho_0 Uu = \rho_0h_0U^2m\cos(kx+mz). \] Hence, the vertical energy flux is \[ pw = \rho_0h_0^2U^3km\cos^2(kx+mz). \] We take the \(x\) direction to be the direction of the background wind, so that \(U>0\), and take \(k>0\). Since \(\cos^2(kx+mz)\geq 0\), the sign of the vertical energy flux is therefore determined by the sign of \(m\). The waves are generated by the mountain, so their energy must propagate upwards, away from the ground. Therefore, requiring the vertical energy flux to be upwards selects \[ m=+\sqrt{\frac{N^2}{U^2}-k^2}. \tag{2.29}\]
Hence, our final solution is \[ \begin{aligned} w &= h_0Uk \cos\!\left(kx+\sqrt{\frac{N^2}{U^2}-k^2}\,z\right),\\ u &= -h_0U\sqrt{\frac{N^2}{U^2}-k^2} \cos\!\left(kx+\sqrt{\frac{N^2}{U^2}-k^2}\,z\right),\\ p &= \rho_0h_0U^2\sqrt{\frac{N^2}{U^2}-k^2} \cos\!\left(kx+\sqrt{\frac{N^2}{U^2}-k^2}\,z\right). \end{aligned} \] The solution has an oscillatory vertical structure rather than decaying with height. As air parcels are advected through this stationary wave pattern, they are displaced alternately upwards and downwards. In sufficiently moist air, the upward motion can cool the air and lead to condensation, producing cloud bands that follow the mountain-wave pattern. Such patterns can be visible in satellite images; two examples are shown in Figure 2.6. Another cool video of these waves created by the Rockies can be found on this page.
In reality, mountains are of course not sinusoidal. However, more general topography can be decomposed into sinusoidal components using a Fourier transform. Since the equations considered here are linear, the wave response to each Fourier component can be calculated separately and the individual solutions superposed. Our simple solution therefore provides a useful building block for determining the waves generated by more general topography.
It is also insightful to consider the case with no stratification. If \(N=0\), Equation 2.25 reduces to \[ \frac{\partial^2 w}{\partial x^2}+\frac{\partial^2 w}{\partial z^2}=0. \] For the same sinusoidal forcing at the ground, the solution that decays as \(z\to\infty\) is \[ w=h_0Uk\,\mathrm{e}^{-kz}\cos(kx). \] The disturbance now has no oscillatory vertical structure and simply decays with height. Without stratification, there is no buoyancy restoring force to support vertically propagating mountain waves.
2.2.5 Hydrostatic approximation
A common approximation in geophysical fluid dynamics is to assume that the horizontal scale is large compared to the vertical scale. In this case, the vertical component of the momentum equation is simply replaced by our hydrostatic balance equation. This is known as the hydrostatic approximation and leads to the following hydrostatic Boussinesq equations
where \(\vb=(u,v)\) is the horizontal velocity and \(\nabla_H=\left(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}\right)\).
Note that the hydrostatic approximation has resulted in the vertical momentum equation being replaced by hydrostatic balance.
The Boussinesq equations (2.20) - (2.22) (and their hydrostatic counterparts (2.30) - (2.33)) are the fundamental equations that are solved numerically in many climate models. Because we observe and measure everything from the rotating Earth, these equations are solved in the rotating frame of reference (which we will come to in Chapter 4). Most climate models use the Hydrostatic equations as they require far fewer computational resources. Hence, because climate models (unlike weather models) are run for very long times (several years or decades), it is worth reducing the computational cost at the expense of making the hydrostatic assumption.
In the geophysical fluids literature, equations (2.30) - (2.33) are often known as the primitive equations. This is somewhat confusing as the non-Hydrostatic Boussinesq equation set is more general. However, the Hydrostatic Boussinesq equations are still more accurate than some other reduced equations such as the shallow water equations and quasigeostrophic equations that we will study in Chapters 3 and 5, respectively.