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DistillWed, 21 Feb 2024 00:00:00 +0000Gravity waves in an ideal fluidValerio Gherardi
https://vgherard.github.io/posts/2024-02-22-gravity-waves-in-an-ideal-fluid
<h2 id="intro">Intro</h2>
<p>We compare two derivations of the stability conditions for
hydrostatic equilibrium of an ideal fluid:</p>
<ul>
<li>A “parcel” argument, that follows the motion of a small particle of
fluid, ignoring the dynamics of the surroundings.</li>
<li>Standard linearization of the ideal fluid equations.</li>
</ul>
<p>The two derivations turn out to give the same answer, but the
intermediate steps in the parcel argument contain some hidden
assumptions, which are clarified in the second approach.</p>
<h2 id="the-parcel-method">The parcel method</h2>
<p>We start with a fluid at rest in a constant gravitational field <span
class="math inline">\(\mathbf g = -g\hat {\mathbf z}\)</span>, and
consider a small portion of fluid initially located at height <span
class="math inline">\(z_0\)</span>. We imagine that this parcel is now
vertically displaced to height <span class="math inline">\(z_1 = z_0 +
\delta z\)</span>, and that no heat is transferred between the parcel
and the surroundings during this process. We further assume that the
pressure inside the parcel rapidly equalizes with the pressure outside
of it (on a time scale much shorter than the one involved in the
displacement). Finally, <em>we assume that the whole process does not
appreciably alter the pressure field <span
class="math inline">\(p\)</span> with respect to its equilibrium
configuration</em>, satisfying <span class="math inline">\(\frac{\text d
p}{\text d z}= -\rho g\)</span>, where <span
class="math inline">\(\rho\)</span> is the fluid’s density at rest. To
anticipate, in the second derivation below, we will see that the last
assumption may actually fail, giving rise to different dynamics than the
one discussed in these Section.</p>
<p>The derivation below follows <span class="citation">(Landau and
Lifshitz 2013)</span>. The parcel’s acceleration in the vertical
direction is given by Newton’s second law:</p>
<p><span class="math display">\[
\rho _\text{p}\ddot {\delta z}=-\rho
_\text{p}g-\frac{\text{d}p}{\text{d}z}=-(\rho
_\text{p}-\rho)g,(\#eq:Newton2)
\]</span> where the equilibrium assumption was used in the second
equality, where <span class="math inline">\(\rho _\text{p}\)</span> is
the parcel’s density, while <span class="math inline">\(\rho\)</span> is
the density of the surroundings evaluated at the parcel’s height <span
class="math inline">\(z_1=z_0 + \delta z\)</span>.</p>
<p>Using the thermodynamic state equation of the fluid, we can express
densities in terms of pressure <span class="math inline">\(p\)</span>
and specific entropy <span class="math inline">\(s\)</span>. For the
fluid density, this means:</p>
<p><span class="math display">\[
\rho = \rho(p(z_1),\,s(z_1)),
\]</span> while for the parcel we have:</p>
<p><span class="math display">\[
\rho _\text{p} = \rho (p(z_1), s(z_0)),
\]</span> due to the fact that the process is adiabatic. Hence,
expanding the right hand side of @ref(eq:Newton2) to first order in
<span class="math inline">\(\delta z\)</span>, we obtain:</p>
<p><span class="math display">\[
\ddot {\delta z} = -\Omega ^2 \delta z,(\#eq:OscArm)
\]</span> where:</p>
<p><span class="math display">\[
\Omega ^2 \equiv -\dfrac{g}{\rho}\left(\frac{\partial \rho }{\partial
s}\right)_p\frac{\text d s}{\text d
z}=-\dfrac{g}{\rho}\left(\frac{\partial \rho }{\partial
s}\right)_p\frac{\text d s}{\text d z},(\#eq:BVFreq)
\]</span> is called the <em>Brunt–Väisälä frequency</em>, or
<em>buoyancy frequency</em> (all quantities in this equation can be
evaluated at <span class="math inline">\(z = z_0\)</span> in the linear
approximation we are considering). Equations @ref(eq:OscArm) imply that,
in order for hydrostatic equilibrium to be stable, we must have <span
class="math inline">\(\Omega ^2 > 0\)</span>, that is:</p>
<p><span class="math display">\[
-\left(\frac{\partial \rho }{\partial s}\right)_p\frac{\text d s}{\text
d z} > 0(\#eq:EquilibriumCond)
\]</span></p>
<p>There are a few alternative ways to express @ref(eq:EquilibriumCond)
<span class="citation">(Landau and Lifshitz 2013)</span>. First of all,
using the Maxwell relation <span
class="math inline">\(\left(\frac{\partial \rho }{\partial
s}\right)_p=\frac{T}{c_p}\left(\frac{\partial \rho }{\partial
T}\right)_p\)</span>, we see that equilibrium requires:</p>
<p><span class="math display">\[
-\left(\frac{\partial \rho }{\partial T}\right)_p\frac{\text d s}{\text
d z}>0.(\#eq:EquilibriumCond1)
\]</span> Moreover, assuming <span
class="math inline">\(\left(\frac{\partial \rho }{\partial
T}\right)_p<0\)</span>, this simplifies to:</p>
<p><span class="math display">\[
\frac{\text d s}{\text d z} >0(\#eq:EquilibriumCond2)
\]</span> Considering <span class="math inline">\(s\)</span> as a
function of <span class="math inline">\(p\)</span> and <span
class="math inline">\(T\)</span>, we have:</p>
<p><span class="math display">\[
\frac{\text d s}{\text d z} = \left(\frac{\partial s}{\partial
T}\right)_p \frac{\text d T}{\text d z}+\left(\frac{\partial s}{\partial
p}\right)_T \frac{\text d p}{\text d z}=c_p \frac{\text d T}{\text d
z}+\left(\frac{\partial V}{\partial T}\right)_p\frac{\text d p}{\text d
z}>0,(\#eq:EquilibriumCond3)
\]</span></p>
<p>where the Maxwell relation <span
class="math inline">\(\left(\frac{\partial s}{\partial
p}\right)_T=\left(\frac{\partial V}{\partial T}\right)_p\)</span> and
the definition of the specific heat at constant pressure <span
class="math inline">\(c_p \equiv \left(\frac{\partial s}{\partial
p}\right)_T\)</span> were used. Finally, using again the equilibrium
condition <span class="math inline">\(\frac{\text d p}{\text d z} = -g
/V\)</span>, we obtain</p>
<p><span class="math display">\[
-\frac{\text d T}{\text d z} < -\frac{\beta Tg}{\rho
c_p},(\#eq:EquilibriumCond4)
\]</span> where <span class="math inline">\(\beta \equiv
-\frac{1}{\rho}\left(\frac{\partial \rho}{\partial T}\right)_p\)</span>
is the thermal expansion coefficient. For an ideal gas, the right hand
side is just <span class="math inline">\(\frac{g}{c_p}\)</span>.</p>
<p>The Brunt–Väisälä oscillation frequency (Eq. @ref(eq:BVFreq)) is
actually correct only in a certain limit, which is best clarified in the
more careful approach, that proceeds from the ideal fluid equations.
Nonetheless, the equilibrium condition <span
class="math inline">\(\Omega ^2 >0\)</span> turns out to be
correct.</p>
<h2 id="linearization-of-the-ideal-fluid-equations">Linearization of the
ideal fluid equations</h2>
<p>In fluid dynamics, our system would be described by the ideal fluid
equations:</p>
<p><span class="math display">\[
\begin{split}
\frac {\text D \mathbf v}{\text D t}&=-\frac{\nabla p}{\rho
}+\mathbf g,
\\
\frac{\text D \rho }{\text D t} &=-\rho (\nabla \cdot \mathbf v),
\\
\frac{\text D s}{\text D t}&=0,
\end{split} (\#eq:IdealFluidEquations)
\]</span> where <span class="math inline">\(\frac{\text D}{\text D t} =
\frac{\partial}{\partial t}+\mathbf v \cdot \nabla\)</span> denotes the
material derivative. The last equation can be exchanged for<a
href="#fn1" class="footnote-ref" id="fnref1"><sup>1</sup></a>:</p>
<p><span class="math display">\[
\frac{\text{D}p}{\text Dt}=c_s^2\frac{\text D \rho}{\text D
t}(\#eq:PressureMatDer)
\]</span></p>
<p>where <span class="math inline">\(c_s^2 \equiv (\frac{\partial
p}{\partial \rho})_s\)</span> is the speed of sound. Denoting by <span
class="math inline">\(p_0\)</span> and <span
class="math inline">\(\rho_0\)</span> the pressure and density field of
the hydrostatic solution, satisfying <span class="math inline">\(\nabla
p _0 = \mathbf g \rho _0\)</span>, we consider a perturbation of the
form:</p>
<p><span class="math display">\[
\mathbf v = \delta \mathbf v,\quad p=p_0+\delta p,\quad
\rho=\rho_0+\delta \rho.(\#eq:DefPerturbation)
\]</span> To linear order in the small quantities <span
class="math inline">\(\delta \mathbf v\)</span>, <span
class="math inline">\(\delta p\)</span> and <span
class="math inline">\(\delta \rho\)</span>, the equations of motion
read:</p>
<p><span class="math display">\[
\begin{split}
0 &=-\frac {\partial \delta \mathbf v}{\partial t}-\frac{\nabla
(\delta p)}{\rho _0}+\mathbf g\frac{\delta \rho }{\rho _0},
\\
0 &=\frac{\partial (\delta \rho) }{\partial t}+\delta\mathbf v \cdot
\nabla \rho _0+\rho_0 (\nabla \cdot \delta \mathbf v) ,
\\
0&=\frac{\partial (\delta p )}{\partial t}+\delta \mathbf v \cdot
\nabla p_0-c_s^2\frac{\partial (\delta \rho )}{\partial t}-c_s^2\mathbf
\delta \mathbf v \cdot \nabla \rho_0.
\end{split} (\#eq:LinSystem1)
\]</span></p>
<p>These equations take a rather simple form if we re-express them in
terms of the mass flux density <span class="math inline">\(\mathbf j =
\rho \mathbf v\)</span>, that is <span class="math inline">\(\delta
\mathbf j = \rho _0 \delta \mathbf v\)</span> to linear order. Before
doing so, we notice that:</p>
<p><span class="math display">\[
\nabla \rho_0=(\frac {\partial \rho}{\partial p})_s\nabla p_0+(\frac
{\partial \rho}{\partial s})_p\nabla s=-(\frac{g}{c_s^2} +\frac{\Omega
^2}{g}) \rho_0 \hat {\mathbf z},(\#eq:GradRho)
\]</span> where <span class="math inline">\(\Omega ^2\)</span> is the
buoyancy frequency defined above (<em>cf.</em> @ref(eq:BVFreq)) and we
assume, consistent with cylindrical symmetry, <span
class="math inline">\(\nabla s\)</span> to lie in the <span
class="math inline">\(\hat {\mathbf z}\)</span><a href="#fn2"
class="footnote-ref" id="fnref2"><sup>2</sup></a>. Putting everything
together, we obtain:</p>
<p><span class="math display">\[
\begin{split}
0 &=-\frac {\partial (\delta \mathbf j)}{\partial t}-\nabla (\delta
p)+\mathbf g\delta \rho,
\\
0 &=\frac{\partial (\delta \rho) }{\partial t}+\nabla \cdot
(\delta\mathbf j) ,
\\
0&=\frac{\partial (\delta p )}{\partial t}+c_s^2\nabla
\cdot (\delta \mathbf j)+\frac{c_s^2\Omega ^2}{g}(\delta \mathbf j)
\cdot \hat {\mathbf z}.
\end{split}(\#eq:LinSystem2)
\]</span></p>
<p>Strictly speaking, the quantities <span
class="math inline">\(c_s^2\)</span> and <span
class="math inline">\(\Omega ^2\)</span> appearing in this equation are
scalar fields with a non-trivial spatial variation. However, assuming
that the spatial scale of the perturbation is much smaller than the
typical scale of variation of <span class="math inline">\(\Omega
^2\)</span> and <span class="math inline">\(c_s^2\)</span>, we can treat
these two numbers as constants. For simplicity, we will work with units
such that <span class="math inline">\(c_s = g = 1\)</span> (this is the
same as using <span class="math inline">\(L=c_s^2g^{-1}\)</span> and
<span class="math inline">\(T = c_sg^{-1}\)</span> as units of time and
length, respectively; the dependence from <span
class="math inline">\(c_s\)</span> and <span
class="math inline">\(g\)</span> can be reintroduced in the final
formulas through dimensional analysis).</p>
<p>The system becomes then a linear system with constant coefficients,
which suggests to search for simple solutions of the form:</p>
<p><span class="math display">\[
\mathbf j = \mathbf u e^{i(\omega t-\mathbf q \cdot \mathbf r)},\quad
\mathbf \delta \rho = \alpha e^{i(\omega t-\mathbf q \cdot \mathbf
r)},\quad\delta p = \beta e^{i(\omega t-\mathbf q \cdot \mathbf r)}.
\]</span></p>
<p>Plugging these into the linearized system, we obtain:</p>
<p><span class="math display">\[
\begin{split}
0 &=-i\omega \mathbf u+i\mathbf q \beta -\hat {\mathbf z}\alpha,
\\
0 &=i\omega \alpha-i\mathbf q \cdot \mathbf u ,
\\
0&=i\omega \beta +\Omega ^2\mathbf u \cdot \hat{\mathbf z}-i\mathbf
q \cdot \mathbf u,
\end{split}(\#eq:LinSystem3)
\]</span></p>
<p>In order to solve these equations, we write:</p>
<p><span class="math display">\[
\mathbf q = q_z \hat {\mathbf z}+q_\perp \hat {\mathbf x}.
\]</span> From the first equation we obtain:</p>
<p><span class="math display">\[
\mathbf u \cdot \hat {\mathbf x} = \frac{\beta q_\perp}{\omega},\quad
\mathbf u \cdot \hat{\mathbf y} = 0, \quad
\mathbf u \cdot \hat {\mathbf z} = \frac{\beta q_z+i\alpha }{\omega}.
\]</span></p>
<p>The second equation then yields:</p>
<p><span class="math display">\[
\dfrac{\alpha}{\beta}=\dfrac {\mathbf q^2}{\omega ^2 -i q_z}
\]</span></p>
<p>Finally, from the third equation we obtain:</p>
<p><span class="math display">\[
0 =\omega ^4 -\omega ^2 [i(1+\Omega ^2) q_z+\mathbf q^2]+q_\perp
^2\Omega ^2
\]</span></p>
<p>We now require <span class="math inline">\(\mathbf q\)</span> to have
an imaginary part <span class="math inline">\(\text {Im}(\mathbf q) =
-\frac{1+\Omega ^2}{2}\hat {\mathbf z}\)</span>, as we rigorously
justify below. Under this assumption, the equation for <span
class="math inline">\(\omega ^2\)</span> has two real roots:</p>
<p><span class="math display">\[
\omega ^2_\pm = \frac {c_s^2 (\mathbf k ^2+\lambda^{-2})}{2}\left(1\pm
\sqrt {1-\dfrac{4\Omega ^2k_\perp^2}{c_s^2 (\mathbf k
^2+\lambda^{-2})^2}}\right),(\#eq:IFFreq)
\]</span> with:</p>
<p><span class="math display">\[
\mathbf k = \text{Re}(\mathbf q),\qquad \lambda^{-1}\equiv \frac{1}{2}(
\frac{g}{c_s^2} +\frac{\Omega ^2}{g})(\#eq:Lambda)
\]</span></p>
<p>(<span class="math inline">\(g\)</span> and <span
class="math inline">\(c_s\)</span> have been reintroduced in these
formulas as explained above).</p>
<p>Before proceeding further, we notice that the frequencies <span
class="math inline">\(\omega _-\)</span> (those from the minus sign
branch in Eq. @ref(eq:IFFreq)) are real if and only if <span
class="math inline">\(\Omega ^2 > 0\)</span>, which is the same as
the equilibrium condition derived from the parcel argument. The actual
frequencies of oscillation are not given by <span
class="math inline">\(\Omega\)</span>, in general.</p>
<h2 id="physical-interpretation">Physical interpretation</h2>
<p>In order to understand the two branches of Eq. @ref(eq:IFFreq), we
start by noticing that, for the whole linearization approach to be
valid, we must have (in natural units <span
class="math inline">\(g=c_s=1\)</span>):</p>
<p><span class="math display">\[
\dfrac{4\Omega ^2k_\perp^2}{(\mathbf k ^2+\lambda^{-2})^2} \ll
1,(\#eq:SmallScaleApprox)
\]</span> This must be the case for the perturbation to be localized in
the <span class="math inline">\(\hat {\mathbf z}\)</span> direction,
which requires <span class="math inline">\(k_z \gg 1\)</span> (notice
that <span class="math inline">\(\Omega, \lambda \sim \mathcal
O(1)\)</span> in natural units).</p>
<p>Assuming @ref(eq:SmallScaleApprox), we can approximate the two roots
in Eq. @ref(eq:IFFreq) as follows:</p>
<p><span class="math display">\[
\omega ^2_+ \approx\mathbf k ^2,\qquad\omega ^2_- \approx \frac{ k_\perp
^2}{k_z^2+k_\perp ^2}\Omega ^2. (\#eq:IFFreqApprox)
\]</span> We also notice that the fluid velocity field satisfies
(without any approximation):</p>
<p><span class="math display">\[
\frac{v_z}{v_\perp} =
\frac{k_z}{k_\perp}\left(\dfrac{1+i\frac{k_\perp^2}{k_z\omega^2}-i\frac{1+\Omega
^2}{2k_z}}{1-i\frac{k_z}{\omega^2}-\frac{1+\Omega ^2}{2\omega^2}}\right)
\]</span> Let us first consider waves associated with <span
class="math inline">\(\omega _+\)</span>, which are essentially sound
waves and for which gravity plays very little role. These have both
phase and group velocity aligned with <span
class="math inline">\(\mathbf k\)</span> and close to <span
class="math inline">\(1\)</span> (the speed of sound), and the fluid
velocity is also in the direction of <span class="math inline">\(\mathbf
k\)</span> (the waves are longitudinal):</p>
<p><span class="math display">\[
\frac{v_z}{v_\perp}\approx \frac{k_z}{k_\perp },
\]</span></p>
<p>In contrast, waves associated with <span class="math inline">\(\omega
_{-}\)</span>, called <em>gravity waves</em>, have vanishing phase and
group velocity in the limit <span class="math inline">\(k_z \to
\infty\)</span>, in general. The material velocity is perpendicular to
<span class="math inline">\(\mathbf k\)</span>:</p>
<p><span class="math display">\[
\frac{v_z}{v_\perp}\approx -\frac{k_\perp}{k_z },
\]</span> The wave frequency depends on the angle <span
class="math inline">\(\theta\)</span> between <span
class="math inline">\(k\)</span> and <span class="math inline">\(\mathbf
g\)</span>, since <span class="math inline">\(\omega _{-}^2 = \sin^2
\theta \cdot \Omega ^2\)</span>. In particular, in the limit of plane
waves in the <span class="math inline">\(\hat {\mathbf z}\)</span>
direction, <em>i.e.</em> <span class="math inline">\(\theta \to
0\)</span>, we have <span class="math inline">\(\omega _{-}^2 \to
0\)</span>, while plane waves orthogonal to gravity <span
class="math inline">\(\theta \to \frac{\pi}{2}\)</span> have frequency
<span class="math inline">\(\omega _{-}^2 \approx \Omega ^2\)</span>.
From the physical point of view, these two limits correspond to the
cases in which the horizontal spatial scale of the perturbation is much
larger/smaller than the vertical scale, respectively.</p>
<p>We realize that the kind of perturbation analysed in the parcel
argument implicitly refers to gravity waves of the second type (with
small horizontal scales). From @ref(eq:IFFreqApprox), we see that the
oscillation frequency coincides with the buoyancy frequency
@ref(eq:BVFreq) only if <span class="math inline">\(k _\perp \gg
k_z\)</span>, that is, if the vertical spatial scale of the perturbation
is much larger than its horizontal scale.</p>
<h4 id="mathematical-details-on-the-wave-solution">Mathematical details
on the wave solution</h4>
<p>In order to justify the procedure used in the derivation of the plane
waves solutions, consider a <em>localized</em> perturbation (say with
compact support) at <span class="math inline">\(t = 0\)</span> and let
<span class="math inline">\(\Psi(t=0)\)</span> denote the vector of
quantities <span class="math inline">\(\delta \rho(t=0)\)</span>, <span
class="math inline">\(\delta p (t=0)\)</span> and <span
class="math inline">\(\delta \mathbf j (t=0)\)</span>. Since <span
class="math inline">\(\Psi(t=0)\)</span> is localized, we can
define:</p>
<p><span class="math display">\[
\widetilde \Psi(\mathbf k,t=0)=\intop \text d ^3 \mathbf r \,e^{i\mathbf
k \cdot \mathbf r} e^{z/\lambda} \Psi(\mathbf r,
t=0)(\#eq:ModifiedFourier)
\]</span> and the inverse of the Fourier transform gives:</p>
<p><span class="math display">\[
\Psi(\mathbf r, t=0) = e^{-z/\lambda}\intop \frac{\text d ^3\mathbf
k}{(2\pi)^3} e^{-i\mathbf k \cdot \mathbf r}\widetilde \Psi(\mathbf
k,t=0).(\#eq:ModifiedFourierInverse)
\]</span> The Fourier components <span class="math inline">\(\widetilde
\Psi (\mathbf k, t)\)</span> for a fixed <span
class="math inline">\(\mathbf k\)</span> satisfy a linear system of
ordinary differential equations, for which we already found four
independent solutions (two for each frequency) in the previous Section.
The fifth solution can be easily verified to correspond to a static,
divergence-less velocity perturbation, with the velocity field
orthogonal to the <span class="math inline">\(\hat {\mathbf z}\)</span>
axis:</p>
<p><span class="math display">\[
\widetilde {\delta \mathbf j}(\mathbf k ,t) = f(\mathbf k) \,{\mathbf k}
\times \mathbf g,\qquad \widetilde {\delta p}=0\qquad\widetilde {\delta
\rho} = 0(\#eq:LastSolution)
\]</span></p>
<p>In momentum space, in the basis provided by the four eigenvectors
with eigenvalues given by @ref(eq:IFFreq), plus this last (static)
solution, time evolution is trivial.</p>
<p>As a parenthetical remark, we notice that if we drop the requirement
of a <em>localized</em> perturbation, we can have additional solutions
that are not covered by the previous remarks. A trivial example is that
of an hydrostatic equations - <span class="math inline">\(\nabla p =
\mathbf g \rho\)</span>, with <span class="math inline">\(\delta \mathbf
j = \mathbf 0\)</span>, all fields being independent of time. These
solutions are clearly not localized, since the pressure field changes
only in the <span class="math inline">\(\hat {\mathbf z }\)</span>
direction. Another example is provided by Lamb waves, that satisfy the
constraint:</p>
<p><span class="math display">\[
\delta \mathbf j \cdot \hat {\mathbf z} = 0
\]</span></p>
<p>and take the general form:</p>
<p><span class="math display">\[
\Psi(\mathbf r,t) = e^{-(\frac{\Omega ^2}{g}-\frac{g}{c_s^2})z}\intop
\frac{\text d ^2 \mathbf k _\perp}{(2\pi)^2} e^{-i\mathbf k _\perp \cdot
\mathbf r} \left[a(\mathbf k _\perp)e^{i kc_st}+b(\mathbf k
_\perp)e^{-i kc_st}\right].(\#eq:LambWaves)
\]</span> These waves are clearly not localized in the <span
class="math inline">\(\hat {\mathbf z}\)</span> direction.</p>
<h2 id="further-problems">Further problems</h2>
<p>This is the point where I felt the algebra was getting a bit too
involved and I left the problem. There are still a few things that it
may be interesting to investigate. In particular, it would be nice to
derive the explicit evolution of a wave packet, say:</p>
<p><span class="math display">\[
\delta \mathbf v (\mathbf r, 0) = \mathbf V
\exp\left[{-\frac{x^2+y^2}{2\sigma _\perp^2}-\frac{z^2}{2\sigma
_z^2}}\right].
\]</span> with vanishing density and pressure perturbation (to simplify
the algebra a little bit). One should compute the “modified” Fourier
transform @ref(eq:ModifiedFourier) and express the coefficients in terms
of the five eigenvectors derived above. Perturbations like this will in
general give rise to a combination of acoustic and longitudinal waves,
depending on the direction of <span class="math inline">\(\mathbf
V\)</span> and on the ratio of vertical and horizontal scales, <span
class="math inline">\(\sigma _z\)</span> and <span
class="math inline">\(\sigma _\perp\)</span>.</p>
<pre class="r distill-force-highlighting-css"><code></code></pre>
<div id="refs" class="references csl-bib-body hanging-indent">
<div id="ref-landau2013fluid" class="csl-entry">
Landau, Lev Davidovich, and Evgenii Mikhailovich Lifshitz. 2013.
<em>Fluid Mechanics: Landau and Lifshitz: Course of Theoretical Physics,
Volume 6</em>. Vol. 6. Elsevier.
</div>
</div>
<div class="footnotes footnotes-end-of-document">
<hr />
<ol>
<li id="fn1"><p>In order to see this, we simply write <span
class="math inline">\(p = p(\rho, s)\)</span> as a function of <span
class="math inline">\(\rho\)</span> and <span
class="math inline">\(s\)</span> and take the material derivative.<a
href="#fnref1" class="footnote-back">↩︎</a></p></li>
<li id="fn2"><p>From the pure mathematical point of view, this is not a
strict consequence of the Eqs. @ref(eq:IdealFluidEquations), which are
in fact consistent with any static density configuration, as long as
<span class="math inline">\(\nabla p = \mathbf g \rho\)</span> is
satisfied. The physical reason is, of course, that we’re neglecting
thermal conductivity, which allows for an arbitrary temperature gradient
to persist forever in the absence of motion.<a href="#fnref2"
class="footnote-back">↩︎</a></p></li>
</ol>
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https://vgherard.github.io/posts/2023-11-15-interpreting-the-likelihood-ratio-cost
Analysis of infinite sample properties and comparison with cross-entropy loss.Forensic ScienceBayesian MethodsInformation TheoryProbability TheoryRhttps://vgherard.github.io/posts/2023-11-15-interpreting-the-likelihood-ratio-costWed, 15 Nov 2023 00:00:00 +0000Conditional ProbabilityValerio Gherardi
https://vgherard.github.io/posts/2023-11-03-conditional-probability
Notes on the formal definition of conditional probability.Probability TheoryMeasure Theoryhttps://vgherard.github.io/posts/2023-11-03-conditional-probabilityFri, 03 Nov 2023 00:00:00 +0000Prefix-free codesValerio Gherardi
https://vgherard.github.io/posts/2023-10-31-prefix-free-codes
Generalities about prefix-free (a.k.a. instantaneous) codesInformation TheoryEntropyProbability Theoryhttps://vgherard.github.io/posts/2023-10-31-prefix-free-codesTue, 31 Oct 2023 00:00:00 +0000AB tests and repeated checksValerio Gherardi
https://vgherard.github.io/posts/2023-07-24-ab-tests-and-repeated-checks
False Positive Rates under repeated checks - a simulation study using R.AB testingSequential Hypothesis TestingFrequentist MethodsStatisticsRhttps://vgherard.github.io/posts/2023-07-24-ab-tests-and-repeated-checksThu, 27 Jul 2023 00:00:00 +0000Testing functional specification in linear regressionValerio Gherardi
https://vgherard.github.io/posts/2023-07-11-testing-functional-specification-in-linear-regression
Some options in R, using the `{lmtest}` package.StatisticsModel MisspecificationRegressionLinear ModelsRhttps://vgherard.github.io/posts/2023-07-11-testing-functional-specification-in-linear-regressionTue, 11 Jul 2023 00:00:00 +0000Sum and ratio of independent random variablesValerio Gherardi
https://vgherard.github.io/posts/2023-06-14-sum-and-ratio-of-independent-random-variables
Sufficient conditions for independence of sum and ratio.MathematicsProbability Theoryhttps://vgherard.github.io/posts/2023-06-14-sum-and-ratio-of-independent-random-variablesWed, 14 Jun 2023 00:00:00 +0000Fisher's Randomization TestValerio Gherardi
https://vgherard.github.io/posts/2023-06-07-fishers-randomization-test
Notes and proofs of basic theoremsStatisticsFrequentist MethodsCausal Inferencehttps://vgherard.github.io/posts/2023-06-07-fishers-randomization-testWed, 07 Jun 2023 00:00:00 +0000p-values and measure theoryValerio Gherardi
https://vgherard.github.io/posts/2023-06-07-p-values-and-measure-theory
Self-reassurance that p-value properties don't depend on regularity
assumptions on the test statistic.Probability TheoryMeasure TheoryFrequentist MethodsStatisticshttps://vgherard.github.io/posts/2023-06-07-p-values-and-measure-theoryWed, 07 Jun 2023 00:00:00 +0000Linear regression with autocorrelated noiseValerio Gherardi
https://vgherard.github.io/posts/2023-05-20-linear-regression-with-autocorrelated-noise
Effects of noise autocorrelation on linear regression. Explicit formulae and a simple simulation.StatisticsRegressionTime SeriesLinear ModelsModel MisspecificationRhttps://vgherard.github.io/posts/2023-05-20-linear-regression-with-autocorrelated-noiseThu, 25 May 2023 00:00:00 +0000Model Misspecification and Linear SandwichesValerio Gherardi
https://vgherard.github.io/posts/2023-05-14-model-misspecification-and-linear-sandwiches
Being wrong in the right way. With R excerpts.StatisticsRegressionLinear ModelsModel MisspecificationRhttps://vgherard.github.io/posts/2023-05-14-model-misspecification-and-linear-sandwichesSun, 14 May 2023 00:00:00 +0000Consistency and bias of OLS estimatorsValerio Gherardi
https://vgherard.github.io/posts/2023-05-12-consistency-and-bias-of-ols-estimators
OLS estimators are consistent but generally biased - here's an example.StatisticsRegressionLinear ModelsModel Misspecificationhttps://vgherard.github.io/posts/2023-05-12-consistency-and-bias-of-ols-estimatorsFri, 12 May 2023 00:00:00 +0000Bayes, Neyman and the Magic Piggy BankValerio Gherardi
https://vgherard.github.io/posts/2023-05-01-magic-piggy-bank
Compares frequentist properties of credible intervals and confidence
intervals in a gambling game involving a magic piggy bank.StatisticsConfidence IntervalsFrequentist MethodsBayesian Methodshttps://vgherard.github.io/posts/2023-05-01-magic-piggy-bankMon, 01 May 2023 00:00:00 +0000Correlation Without CausationValerio Gherardi
https://vgherard.github.io/posts/2023-03-10-correlation-without-causation
*Cum hoc ergo propter hoc*Statisticshttps://vgherard.github.io/posts/2023-03-10-correlation-without-causationThu, 30 Mar 2023 00:00:00 +0000How to get away with selection. Part II: Mathematical FrameworkValerio Gherardi
https://vgherard.github.io/posts/2022-11-07-posi-2
Mathematicals details on Selective Inference, model misspecification and coverage guarantees.StatisticsSelective InferenceModel Misspecificationhttps://vgherard.github.io/posts/2022-11-07-posi-2Fri, 25 Nov 2022 00:00:00 +0000How to get away with selection. Part I: IntroductionValerio Gherardi
https://vgherard.github.io/posts/2022-10-18-posi
Introducing the problem of Selective Inference, illustrated through a simple simulation in R.StatisticsSelective InferenceRhttps://vgherard.github.io/posts/2022-10-18-posiMon, 14 Nov 2022 00:00:00 +0000kgrams v0.1.2 on CRANValerio Gherardi
https://vgherard.github.io/posts/2021-11-13-kgrams-v012-released
kgrams: Classical k-gram Language Models in R.Natural Language ProcessingRhttps://vgherard.github.io/posts/2021-11-13-kgrams-v012-releasedSat, 13 Nov 2021 00:00:00 +0000R Client for R-universe APIsValerio Gherardi
https://vgherard.github.io/posts/2021-07-25-r-client-for-r-universe-apis
{runi}, an R package to interact with R-universe repository APIsRhttps://vgherard.github.io/posts/2021-07-25-r-client-for-r-universe-apisSun, 25 Jul 2021 00:00:00 +0000Automatic resumes of your R-developer portfolio from your R-UniverseValerio Gherardi
https://vgherard.github.io/posts/2021-07-21-automatically-resume-your-r-package-portfolio-using-the-r-universe-api
Create automatic resumes of your R packages using the R-Universe API.Rhttps://vgherard.github.io/posts/2021-07-21-automatically-resume-your-r-package-portfolio-using-the-r-universe-apiWed, 21 Jul 2021 00:00:00 +0000{r2r} now on CRANValerio Gherardi
https://vgherard.github.io/posts/2021-07-06-r2r
Introducing {r2r}, an R implementation of hash tables.Data StructuresRhttps://vgherard.github.io/posts/2021-07-06-r2rTue, 06 Jul 2021 00:00:00 +0000Test postValerio Gherardi
https://vgherard.github.io/posts/2021-07-06-test-post
A short description of the post.Otherhttps://vgherard.github.io/posts/2021-07-06-test-postTue, 06 Jul 2021 00:00:00 +0000