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\begin{document}
\title{\sffamily Comments and Errata on Atmospheric and Oceanic Fluid Dynamics (1st Printing)}
\author{\sffamily Geoffrey K. Vallis}
%\date{ }
\maketitle
%

\noindent 


The following errata and comments apply to the \emph{first printing} of the book. Many of the errors have been corrected in the second printing, although a few remain. If you have the second (or later) printing of the book, available from October 2007, please see the corresponding errata sheet. (The later printings have `Reprinted 2007' or similar on the copyright page.) Errors that occur in both printings are (or should be) listed in both errata sheets.

Most of the errors should not, I hope, cause too much frustration to either the careful reader (who will figure them out independently) or the casual reader (who won't notice them), but those marked with a black diamond may cause a little head-scratching or may need the reader to work a little. These are collected at the head of the list, and are followed by the typos in page order. Unless stated, none of the corrections affect the subsequent arguments or derivations in the book. If you find other errors, or if you think something is poorly explained, please contact the author at `gkv-at-princeton-dot-edu'. 


\begin{enumerate}

\item \star \label{cancel} Pages 156 and 158. The cancellation of certain boundary terms is poorly explained. However, the results are correct. 
% Email me if you wish for a clarification sooner rather than later.

\item \star \label{informal} Page 258. This explanation (the informal mechanism) is a little brief and may be hard to follow, and (6.48) is not transparent without more algebra. A clearer version may be provided in the second edition. 

\item \star \label{eadyschem} Page 262, Fig.\ 6.9. The labels {\sf B} and {\sf C} are interchanged. 
% Note 'rho' instead of $\rho$. 

\item \star \label{simmhosk} Page 399, fig.\ 9.9 caption.  Potential energy and kinetic energy are transposed. Should read: ``\ldots Initially baroclinic processes dominate, with conversions from zonal to eddy APE and then eddy APE to eddy kinetic energy, followed by the barotropic conversion of eddy kinetic to zonal kinetic energy\ldots"

\item \star \label{channel} Page 266. Above (6.78), the channel extends from $0$ to $L$, not $\pm L/2$. Relatedly, on page 268, above (6.91), the most unstable wave is that with the gravest meridional scale, and so the smallest possible meridional wavenumber $l$. This is not exactly $l=0$ except in doubly periodic geometry, but for a wide channel $l$ is very small.

\item \star Pages 426--436. In a few places, `across gradient' is written where `across iso-surfaces' (or `along gradient') is meant, and  `diagradient' should be replaced by `along-gradient' (usually downgradient) or `diffusive' (`dia' means \emph{across}). Page 426, below (10.82), should read: `The flux has a component across the isosurfaces of $\phi$, called the diffusive flux, and a component along the iso-surfaces, called the skew flux'. Page 428, l3--4, `along-gradient' should be `along isosurfaces' (or `perpendicular to the gradient'). Page 429, section 10.6.4, first bullet should read `\ldots across isosurfaces, the diffusive flux, and a component along isosurfaces, the skew flux.'  Page 436, item (i) should read `\ldots across iso-surfaces of buoyancy' instead of across its gradient. Duh.

\item \star Page 434, \eq (10.135). The inequality needs further explanation, especially as $\ppp by $ may be positive or negative.  The reader may ignore it, and just read on.

\item \star Page 525, \eq (PV.1). The term $-\ppp{\obar{v_i q_i}}y$ should be just $\obar{v'_i q'_i}$, as in (12.67). This typo should be obvious to those who have read the text, but it is still confusing. Relatedly, the term on the lhs of (PV.4) should have a minus sign.

\end{enumerate}

\bigskip

\begin{enumerate}

\item Page 5, \eq (1.10).  Given the  convention used in the book, the subscripts on $b$ should be superscripts (i.e., $b^x$ not $b_x$), because they denote components of a vector and not derivatives. 

\item Page 21, first line of last paragraph. $\bppp h t _p$ should be $\bppp hT_p$

\item Page 24, \eq (F.2). Second term in left equation should be plus, not minus. See \eq (1.93).

\item Page 27, expression for $Q[\rho]$ on line 3. The second term on the right-hand side should be $+ (\ppp \rho S)_{\eta,p}$ and not $-(\ppp \rho S)_{\rho,p}$.

\item Page 29, equation on last line. $\d p$ should be $\d \alpha$ in middle term on rhs.

\item Page 31, line below (1.139). Should be: $\theta = \theta(\eta, S, \pref) = \theta (T, S, p)$; i.e., $p$ not $\pref$ in last term.

\item Page 34, line before (1.148). Should be $\alpha  = (\ppp G p)_{T,S}$ and not
$\alpha  = (\ppp G p)_{T,p}$.

\item Page 35, Fig.\ 1.4 caption, line 3. Replace 13.36\dgc by 10.14\dgc. 


\item Page 64, \eq (2.62). There should be no $\rho$ in the first fraction on the right-hand side.  

\item Page 64, \eq (2.68). The term $\ppp{(\rho m w)} z$ should be $r^{-2}\ppp{(\rho m w r^2)} r $.

\item Page 70, expression following (2.98). Should be minus sign in front of $\cp/(T 
\ro\beta_T)$. [Note the use of the equation of state (1.59).] 

\item Page 74, \eq (2.122). For notational consistency, $\rho'$ should be $\delta \rho$, and similarly in the sentence immediately following.

\item Pages 81--82, Eqs. (2.167) and (2.174). On right-hand sides $-b$ should be $+b$ and, for more clarity, $\del \phi$ should be $\del_z\phi$ (it is a horizontal derivative).

\item Page 86, line 3. The factor of $\sin \theta$ should be removed, but (2.187) is nevertheless correct.

\item Page 91, \eq (2.213). Scaling for $t$ should be $L/U$, not $U/L$.

\item Page 119, problem 2.14. The pressure term, $-\ro^{-1} \grad p$, is inadvertently omitted from the right-hand side of the first equation. 

\item Page 130, \eq (3.33). Extraneous minus sign on rhs.

\item Page 136--137, section 3.6.1. It should be probably be stated explicitly that we are considering the $f=0$ case, especially as (3.68) only follows from (3.8) if $f=0$. The rotating case follows on page 138. 

\item Page 139, section 3.6.2. The derivation given is for the flat-bottomed case, which is not stated explicitly. As an exercise, the reader should perform a similar calculation with topography, and show that $E = (h u^2/2 + g h^2/2 + g \eta_b h)$ and $\Fb = \ub( E + g h^2/2)$.  

\item Page 142. The deformation radius, $L_d$, is not defined. It is $L_d \equiv \sqrt{gH}/f$. (In the continuously stratified case, it is $NH/f$.)

\item Page 147, line before (3.134). In fact, the condition that is imposed is that the \emph{derivative} of streamfunction (i.e., the velocity) goes to zero, not streamfunction itself. 

\item Page 150, (3.149) and (3.150). Factor of $\er^{\ir (kx + ly)}$ is missing in all the integrands. 

\item Page 154. \item Page 154, eq. (3.171). $\DDD \vb$ should be $\DDD \ub$. Also, 
Section 3.9.3 is hard to follow as written.

\item Page 156, \eq (3.178). The two right-hand sides should each have a minus sign before the integral signs. \\
\Eq (3.179). Factor of $1/2$ missing, and the minus sign should be deleted. The factor of 1/2 should be propagated through to (3.183). 

\item Page 160. Formatting error in problem 3.3. It should read $\div[][\vb(E + gh^2/2)]$ and not $\div[\vb(E + gh^2/2)]$. 

\item Page 175. Just after (4.60), should be $\vb_r = \vb_a - \Omb \cross \rb$ (no factor of 2).

\item Page 185, \eq (4.101). Second $v$ should be a $u$. \\ Regarding (4.102), the second equation is not clear since we are comparing two infinitesimals. Best to simply ignore it, as the derivation still goes through. A better explanation will be provided in the second edition.  

\item Page 186, \eq (4.109). Should read $\alpha^{-2} \gg 1$ and not $\alpha^2 \ll 1$. Also, line before (4.113), $\Omb = f \kb$ should be $2\Omb = f \kb$.

\item Page 196, \eq (5.1). Should read $(u, v) \sim U$, not $(u,v) \sim L$.

\item Page 204, \eq (5.43). Should be $\ub = \kb \cross \del \psi$ (no minus sign). 

\item Page 208, \eq (5.59). The term $\what \fo k\cross\hat{\ub}_1$ has an extraneous minus sign. \\
\Eq (5.61). Missing subscript 0 on second $\ubhat$, and second $t$ should be $\that$. Also, missing $\fhat_0$ in (5.62), although its value is unity.

\item Page 212. After (5.81), there is a $g$ missing in the definition of $g'$.

\item Page 213, eqs.\ (5.85a) and (5.88a). Should be a minus sign before last terms on rhs in each equation, and $H_2$ should be $H_1$ in last term in (5.88a). (The terms are subsequently dropped).  

\item Page 219, \eq (5.122). $\ppp \psi z$ should be multiplied by $\fo$.

\item Page 221, \eq(5.131). $\psi$ should be $\ppp \psi Z$ in advection term.

\item Page 224. In (5.140) we should have $\ppp[2] \psi z$ and not $\ppp \psi z$, and in (5.141) $\what q \equiv \wtilde Q$. 

\item Page 225, \eq (5.151d) should read: 
 $ \sigma' =  - \fo \ppp{}\theta \left(\trho \theta \, \ppp{\psi'} \theta \right)$  .

\item Page 227, \eq (5.159). $H$ should be $H^2$.

\item Page 228. Equations (5.167) have typos and should be ignored.  In (5.168) $\psihat \equiv (\psi_1 -\psi_2)/2$. Also, in (5.170), $\int \dxb \equiv \int \dA$. 

\item Page 232. End of line 2, eastwards should be westwards.

\item Page 237--238 on phase speed. Some sources define the phase
velocity to be given by $\cb_p \equiv \omega \kbi / K^2$, where $\kbi$
is the wavevector. The components of the phase velocity are then given
by $c^x_p = \omega k^x/K^2$, etc. Defined this way, the phase velocity
is a true velocity. However, its components do \emph{not} represent the
speed at which wavecrests travel along the coordinate axes. This
definition is not common, but be aware of it. Also, at the bottom of
page 240, the explanation of group velocity is rather terse and note
that ${\omega'} = \omega(k+k') - \omega(k)$.

\item Page 239, \eq (5.225). Last term should have minus sign before it.

%\item Page 252, \eq (6.22). Remove minus sign on rhs.  

\item Page 256, caption to Fig.\ 6.5. $\sigma = -i k c_i$ not $\sigma = kc_i$.

%\item Page 258, \eq (6.47). Both equations are missing a factor $\exp(-2ka) = \exp(-1)$. 

\item Page 259 \eq (6.53) and (6.54). The terms $v \zeta$ should be multiplied by 2. \\ Eq.\ (6.55), the two plus signs should both be minus signs, as in (7.17).


\item Page 263, \eq (6.61). $z_A$ instead of $z_B$ in fourth term in second expression. 

\item Page 265, after (6.74). Should be `the integral must vanish', or alternatively (and implying) `the integrand must vanish at some point'. 

% \item Page 264. Double `for' at beginning of section 6.4.3.
% \item Page 265, \eq (6.73). Missing tilde on first term.
% \item Page 266, l2. Where $H$ \emph{is} the domain depth. (Missing is.)

\item Page 266. It is the background PV \emph{gradient} that is zero in the Eady problem. $Q$ in (6.75), which is zero, is the contribution to the basic state PV from the flow and from $\beta y$; this is sometimes called the `QG PV', and does not include the $f_0$ term. 


% \item Page 268, figure caption. Better to use $U$ instead of $u$.

% \item Page 272, above (6.106). Double `basic'. The word 'gradient' should be omitted. 

\item Page 274, l--2. $\beta \neq 0$, not $\beta = 0$. 

\item Page 277, surrounding (6.128). $K$ instead of $k$, so the unstable interval is $\sqrt{\beta/2U} < K < k_d$.

\item Page 275, first sentence of second section. No hats on variables.

% \item Page 278, 'of with' under (6.129). Also, comment explicitly that \y variation is ignored. Define `neutral waves'. 

\item Page 279. Better to have primes on the perturbation variables in (6.135)--(6.137). 

\item Page 280, \eq (6.141). $c_T$ and $c_B$ are interchanged.

% \item Page 281, l--4. Should be, `$b$ the role of $u$.'

\item Page 284, \eq (6.161) should read: $(\ppp{}t + U \ppp {}  x) q' + (\ppp{\psi'} x)( \ppp Q y) = 0 $ and not $(\ppp{}t + U \ppp u x) q' + (\ppp{\psi'}x) ( \ppp Q y )= 0 $. (Spot the difference.)

\item Page 285, \eq (6.165). $\psi$ should be $\psi'$ on lhs, and four lines above. 
 
\item Page 296. Just before (7.5), `If $J$ is also zero' should be `if $H$ is also zero'. Just after (7.5), `$D$ and $J$ are zero' should be `$D$ and $H$ are zero'. Just below (7.12) on p.\ 297, `where $J'$ is' should be `where $H'$ is'.

\item Page 298, first sentence of section 7.2. `Eddy flux of potential may be' should be `Eddy flux of potential vorticity may be'.

\item Page 300, \eq (7.33). $k$ and $l$ are transposed in the expressions for $\util$ and $\vtil$. We should have:
     $  \util  =  - \ir l \psitil,  \quad \vtil = \ir k \psitil$. 

\item Page 303, l 7. $p$ should be $m$, twice. 

\item Page 304, \eq (7.60b). $w$ should be $\wbar$. 

\item Page 308. In (7.79), the first $\sigmabar$ should be $\hbarr$. Better, 
$\obar[b]v \equiv  \obar[b]{v \sigma}/\sigmabar =  \obar[b] v + \obar[b]{v'\sigma'} /\sigmabar  $.

\item Page 310, second bullet. $\bm E$ should be $\Fbc$.

\item Page 311, \eq (7.93). $\phi$ in first term should be $\varphibar$.

\item Page 313, \eq (7.106). Should be $\Fb[\varphi]$ not $\Fb[\varphibar]$.

\item Page 314, \eq (7.112c). Middle equation should have minus sign on rhs, and in last equation $\vb'$ should be $\ub'$ (the horizontal eddy velocity).

% \item Page 319, line after (7.129). $\eta$ should be $\eta'$. 

\item Page 319, eqs.\ (7.130) and (7.131). Missing factor of $\ro$, and, in (7.131) only, $\fo$.

\item Page 320, in paragraph above (7.138). First and last references to (7.136) should be to (7.135).

\item Page 324, \eq (7.141). Subscripts $1$ should be subscripts $i$. 

% \item Page 326, line above (7.152). Replace first minus sign by equals sign, giving $\fo \ppp \ubar z = - \ppp \bybar y$.

\item Page 327, \eq (7.157). No minus sign on rhs.

\item Page 330, first equation in box (7.812). Extraneous $s$ on lhs, next to $\what H$.

\item Page 339, \eq (8.8). Should be $\zeta$ and not $\psi$ in viscous term. 

% \item Page 339, \eq (8.10). Wrong sign on both terms on rhs.

\item Page 340. $1/L^2$ should be $L^2$ in (8.11), and $k^4$ should be $k^2$ in (8.12).

\item page 343. $\wtilde u(\kbi)$ should be $\wtilde u(\kbi,t)$. 

\item Page 345. In fig.\ 8.3, $E(k)$ should be $\Eta(k)$.

\item Page 352 and 353, \eq (8.48) and \eq (8.53). $\Ehat$ should be $\Ebar$, and $\Zhat$ should be $\Zbar$, respectively.

\item Page 352, \eq (8.49).  $\Zeta(q)$ is not the same function as the $\Zeta(k)$ of (8.44), but it is still such that the enstrophy is $\int \Zeta(q) \, \d q$. 

\item Page 355, \eq (8.62a). The integral $\int \dxb$ is the same as $\int \d A$, and the viscous term should have a minus sign. In (8.63)--(8.65), for notational consistency the energy spectrum should be written $\Eta(k)$ and not $E(k)$.

\item Page 369, \eq (8.105). $\chi$ should be $\chi'$ on right-hand side. ($\chi$ is a constant.) And replace `by $\eta^{-1/3}$' by `is $\eta^{-1/3}$' in
the second sentence of the paragraph above (8.107). 

\item Page 380, caption and inset to Fig.\ 9.1. $\veps$ should be $\veps^{1/3}$. 

\item Page 386, \eq (9.24b). $a$ and $b$ are transposed on second term on rhs of definition of Jacobian. Also, line before (9.27), second `barotropic' should be `baroclinic'.

\item Page 389. 1st text line should read $|\del^2| \sim
k^2 \ll \kd^2$ and not $\del^2 \sim
\kd^2 \ll k^2$.

\item Page 392, \eq (9.48). $D[\tau]$ should be multiplied by $-k_d^{-2}$, but the term is not used.

\item Page 404 and 725. Holloway and Hendershott (1986). Year should be (1977).

\item Page 417. The symbols $K_{\textit{turb}}$ and $\K_{\textit{turb}}$ are both used for the same quantity, namely the turbulent diffusivity.

\item Page 428, line after (10.100). `Divergence' should be `gradient', twice.

\item Page 434, \eq (10.132): $\ubar$ should be $\ubbar$ in first term. And line after (10.132): `Zonally uniform, small amplitude wave' should read `Zonally uniform basic state and small amplitude wave'. In (10.133), $u'$ should be $v'$. Two lines after (10.134), reference to (10.133) should be omitted.

\item Page 437, \eq (10.146). Subscripts on $s$ should be superscripts, as they denote components and not derivatives. 

\item Page 440, \eq (10.156). The $\ub'$ should be outside of the derivative in the third order term, but this term is promptly neglected. 

\item Page 443, \eq (10.175). $\obar{\vb' \sigma'}$ should be $\obar{\ub' \sigma'}$ in third term.
 
\item Page 456, \eq. (11.3). The signs on the right-hand sides of both equations such be flipped in order to be consistent with figure (11.3) and \eq (11.68). Note, though, that there is no a priori correct definition of the sign of a streamfunction.

\item Page 458 \eq (11.4). Need a factor of $1/a$ on first term of rhs. Also, $\partial_y$ should be $\partial_\vrt$ on the following line; alternatively, delete factor of $a^{-1}$.

\item Page 460, \eq (11.11). Missing factor $-1/g$ on rhs.

\item Page 468, item \textit{(ii)} and (11.40). Note that we are using an \textit{equatorial} beta plane, on which $\fo=0$. Thus, $f=\fo + \beta y = \beta y$.

\item Page  471, \eq (11.54). For notational consistency with what follows, $\delta \theta$ should be $\theta$. 

\item Page 471. Line above (11.57), should be $m(0) = \Omega a^2 \cos^2 \vrt$ (i.e., $\vrt$ and not $\vrt_1$), as $u=0$ at the surface. And (11.57) and (11.58) are missing factors of $-\theta_0$ on rhs.

\item Page 474, $\cos \vrt$ should be $\cos^2 \vrt$ in text at end of second paragraph.

\item Page 476, \eq (11.64); page 480, \eq (11.73);  page 481, \eq(11.75). The eddy flux term needs a factor of $1/a$.

\item Page 476, \eq (11.66). $w$ should be $\wbar$.

 \item Page 494, \eq (12.34). Factor of 1/2 missing on right-hand sides. See, e.g., (12.27). 

 \item Page 495. The statement `illustrating the group velocity property' is opaque and best ignored. 

\item Page 504. Line after (12.61b), extraneous $-\nabla$ between $g'$ and $\eta$. Also, $g' = g (\rho_2-\rho_1)/\ro$, and not $g' = g (\rho_1-\rho_2)/\ro$. Relatedly, (12.64) should have a minus sign on rhs. (It is technically correct as written, because of the way $g'$ is defined.) Finally, after (12.63), note that $\phi_T = \phi_1$, and its introduction is unnecessary.

\item Page 505, \eq(12.71). $h_i$ should be $\hbarr_i$. 

\item Page 506, \eq (12.77). $v_1$ on rhs should be $v_2$. (Eq.\ (12.79) is correct.) 

\item Page 507. \Eq (12.83), second term on rhs should have plus sign, not minus sign.

\item Page 508. \Eq (12.85), $q$ should be $\qbar$, twice.

\item Page 511. Sentence above (12.89), polewards and equatorwards should be reversed. And `inequality' should be `equality' in the line following (12.89).

\item Page 513. $H$ in (12.95) is not defined nearby, and is $H = H_1 + H_2$. Also, the frictional term in (12.95) and (12.96) has the wrong sign, and the factor of 2 in the first term of (12.96) is spurious. The terms are not used elsewhere.

\item Page 514. In (12.98) $\eta$ should be $\etabar$, and in (12.99) $q$ should be $\qbar$.

\item Page 516. In (12.103), factor of 1/2 should be inside square brackets.

\item Page 518. Line 2: decreases should be increases. Line 7, westward should be westerly. 

\item Page 531, fifth line after (12.119). Tropopause should be troposphere. Likewise on page 533, third line after (12.123) 

\item Page 533, fourth line after (12.120), second subscript should be $s$, not $i$. 

\item Page 536--537. The appendix discusses the computation of the EP fluxes in log-pressure coordinates. However, the computations were actually carried out in pressure coordinates, with the scaling as indicated at the bottom of page 537, and then the results transformed to log-pressure coordinates for plotting purposes only. 

\item Page 539, endnote 12. The reference to page 12.6.3 should be to page 532, or to section 12.6.3.

\item Page 543. Missing prime on $v$ in (13.7) and (13.10).
 
\item Page 547. The factor $1/k_s$ should be omitted in (13.18).

\item Page 551, below (13.34). Would be clearer if it read: `\ldots, and so \emph{short} waves are trapped by westerly flow.'
%All waves are trapped by easterly flow.'

\item Page 555, \eq (13.50), and later in (13.66). First term, $\ppp{\psi} t$ should be $\ppp{ (\pppp \psi z)} t$.  

%\item Page 556. In (13.55), last term on lhs should be multiplied by $K^2$,   but this term is dropped anyway.

\item Page 556. In  (13.60a), minus sign after $\ubar$ should be plus sign [as in (5.186b)].

\item Page 555--559. We write the topography as $h_b(x,y)= \Re [ \wtilde{h}_b \sin ly \e^{\ir kx} ]$, and then in (13.55), (13.58), (13.65), (T.1a,b) and the rhs of (13.70), $h_b$ should be replaced by $\wtilde h_b$.

\item Page 558. In (13.65), $im$ should be $-im$. In the paragraph following,
   it is $\rho_R |\psitil|^2$, not the amplitude of $\psitil$ itself, that is independent of height in the last case. 

\item Page 561, \eq (13.76b). The second term should be multiplied by $\fo$. 

\item Page 574. The right-hand sides of (13.93) and (13.94) should be multiplied by $-1$.

\item Page 586, eqs.\ (14.5) and (14.14). The quantity $W_E$ is not quite the same as the vertical velocity due to Ekman pumping, although the notation might imply it. The Ekman pumping velocity, $W_{\textit{Ek}}$, is given by (2.296), and so $W_{\textit{Ek}} \approx \curlz (\taub/f)$, whereas $W_E = \curlz \taub$.  
 


\end{enumerate}

\noindent \textit{Acknowledgment:} I am grateful to all the readers who have sent in comments. I would particularly like to acknowledge Roger Berlind for his exceptionally close reading of the entire book.

\end{document}

% Finis
