= Solution
On a sphere of radius $R$ rotating at angular speed $\Omega$, the <Coriolis parameter> is $f=2\Omega\sin\phi$. Near latitude $\phi_0$, write $y=R(\phi-\phi_0)$, use locally eastward and northward Cartesian coordinates, and expand
$$
\boxed{f=f_0+\beta y,\qquad f_0=2\Omega\sin\phi_0,\qquad \beta=\frac{2\Omega\cos\phi_0}{R}.}
$$
The <beta plane> retains the first northward variation of <planetary vorticity> while neglecting higher latitude dependence and metric curvature. A midlatitude local calculation assumes $L/R\ll1$ and $|\beta|L/|f_0|\ll1$; near the equator the distinct equatorial <beta plane> has $f_0=0$, so the midlatitude low-frequency reduction used below does not apply.
For a homogeneous shallow layer, the <hydrostatic approximation> gives $p=p_{\rm atm}+\rho g(\eta-z)$ and hence $p_x/\rho=g\eta_x$, $p_y/\rho=g\eta_y$, independently of depth. The approximation follows from the small aspect ratio $H/L$ and neglect of vertical acceleration. Differentiate the linear horizontal momentum equations in $z$. Their depth derivatives obey
$$
(u_z)_t-fv_z=0,\qquad (v_z)_t+fu_z=0.
$$
Starting from rest gives $u_z=v_z=0$ initially, and the unique solution remains zero. This establishes <depth independence of hydrostatic shallow-water flow>; an arbitrary pre-existing shear would not be removed merely by taking the <hydrostatic approximation>. Integrating the <continuity equation> between the rigid bottom and the moving surface gives $\eta_t+H(u_x+v_y)=0$ at linear order. With the <depth-integrated shallow-water transports> $U=Hu$, $V=Hv$, one obtains
$$
\boxed{U_t-fV=-c^2\eta_x,\qquad V_t+fU=-c^2\eta_y,\qquad \eta_t+U_x+V_y=0,\qquad c^2=gH.}
$$
It is useful to make the coefficient approximation in the height reduction explicit. Put $\mathcal F_y=\partial_t^2+f(y)^2$. Differentiating the two momentum equations in time and eliminating the other transport gives
$$
\mathcal F_yU=-c^2(\eta_{xt}+f\eta_y),\qquad \mathcal F_yV=-c^2(\eta_{yt}-f\eta_x).
$$
Since $\partial_y(\mathcal F_yV)=\mathcal F_yV_y+2f\beta V$, taking their horizontal <divergence> and using the <continuity equation> gives
$$
\left[\nabla_h^2\eta-\frac{\mathcal F_y\eta}{c^2}\right]_t=\beta\eta_x-\frac{2f\beta}{c^2}V.
$$
Applying $\mathcal F_y$ proves the exact <variable-Coriolis shallow-water height equation>
$$
\boxed{\mathcal F_y\left[\nabla_h^2\eta-\frac{\mathcal F_y\eta}{c^2}\right]_t=\beta(\eta_{xtt}+2f\eta_{yt}-f^2\eta_x).}
$$
At the reference latitude, or after the usual local freezing of undifferentiated $f$ factors to $f_0$, this gives the height relation written with $\mathcal F_0=\partial_t^2+f_0^2$. With $f=f_0+\beta y$ over a finite region, that constant-coefficient version is a local approximation, not an exact identity. Keeping $\mathcal F_y$ as above avoids silently commuting a variable <Coriolis parameter> through a spatial derivative.
For the slow <Rossby wave> branch, take $|\omega|/|f_0|\ll1$, approximate $\mathcal F_0$ by $f_0^2$, and discard the two time-derivative terms on the right compared with $-f_0^2\eta_x$. This is the regular midlatitude long-time ordering, with nondegenerate zonal variation. If $a=f_0/c$, the result is
$$
\boxed{[\nabla_h^2\eta-a^2\eta]_t=-\beta\eta_x.}
$$
The printed low-frequency equation has the opposite right-hand sign. The minus sign follows directly from the preceding eliminated equation and is also the sign needed for the printed isofrequency-circle centre. An independent check uses <geostrophic balance>: $\psi=g\eta/f_0$, $(u,v)=(-\psi_y,\psi_x)$, and the <shallow-water quasi-geostrophic potential vorticity> is $\nabla_h^2\psi-a^2\psi+\beta y$. Linear <potential-vorticity conservation> gives $(\nabla_h^2\psi-a^2\psi)_t+\beta\psi_x=0$.
For $\eta=Ae^{i(kx+ly-\omega t)}$, the <shallow-water Rossby-wave dispersion relation> is
$$
\boxed{\omega=-\frac{\beta k}{k^2+l^2+a^2}.}
$$
For fixed $l$ and $\beta>0$, it is odd in $k$, zero at $k=0$, negative for $k>0$, and tends to zero from below as $k\to+\infty$. Its minimum occurs at $k=\sqrt{l^2+a^2}$ with $\omega_{\min}=-\beta/[2\sqrt{l^2+a^2}]$. Thus both long and short waves have small frequency. The zonal <phase velocity> is westward, $\omega/k<0$, whereas the zonal <group velocity> is
$$
c_{gx}=\frac{\partial\omega}{\partial k}=\frac{\beta(k^2-l^2-a^2)}{(k^2+l^2+a^2)^2}.
$$
It changes sign at the frequency minimum.
At fixed nonzero frequency, completing the square yields the <Rossby-wave isofrequency circle>
$$
\boxed{\left(k+\frac{\beta}{2\omega}\right)^2+l^2=\frac{\beta^2}{4\omega^2}-a^2.}
$$
The radius is real only if $|\omega|\leq|\beta|/(2|a|)$. For $\beta>0$, the branch with $k>0$ has $\omega<0$ and a circle centred on the positive $k$-axis. If instead the printed plus-sign wave equation is taken literally with this same Fourier convention, its dispersion is $\omega=+\beta k/(k^2+l^2+a^2)$ and its circle is centred at $(+\beta/(2\omega),0)$. These two conventions cannot be mixed.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-79-rossby-dispersion.png]
{title=Rossby-wave frequency curves and a constant-frequency wavenumber circle showing a westward-group incident wave and an eastward-group reflected wave at a meridional wall}
{height=500}
For <reflection of a Rossby wave at a meridional wall>, the stationary wall preserves frequency, and its translation invariance in $y$ preserves the tangential wavenumber. Thus $\omega_r=\omega_i$, $l_r=l_i$. Both $k$ values solve
$$
\omega_i k^2+\beta k+\omega_i(l_i^2+a^2)=0.
$$
Their sum and product give
$$
\boxed{k_r=-\frac{\beta}{\omega_i}-k_i=\frac{l_i^2+a^2}{k_i}.}
$$
An incident wave in $x>0$ travels toward the wall in <group velocity>, not necessarily in <phase velocity>. For $\beta>0$, $0<k_i<\sqrt{l_i^2+a^2}$ has $c_{gx,i}<0$, while its partner $k_r>\sqrt{l_i^2+a^2}$ has $c_{gx,r}>0$. The double-root case has zero normal <group velocity> and does not describe a wave packet incident on the wall.
At leading <quasi-geostrophic approximation>, the <impermeability condition> is $u=-g\eta_y/f_0=0$. For $l_i\ne0$, the boundary condition gives
$$
-i\frac{g l_i}{f_0}(A_i+A_r)e^{i(l_i y-\omega_i t)}=0,\qquad \boxed{A_r=-A_i,\quad |A_r|=|A_i|.}
$$
The reflected height therefore has equal amplitude and a phase change of $\pi$. Equality here concerns the height or <quasi-geostrophic streamfunction> amplitudes in the reduced model. At the degenerate $l_i=0$, geostrophic no-normal-flow is automatically satisfied and alone does not determine their ratio; the usual homogeneous wall-streamfunction condition supplies $A_r=-A_i$ if imposed. Retaining the small ageostrophic transport changes the boundary condition to
$$
\frac{A_r}{A_i}=-\frac{\omega_i k_i+i f_0l_i}{\omega_i k_r+i f_0l_i},
$$
which tends to $-1$ in the regular low-frequency ordering but is not generally of unit modulus. In particular, $l_i=0$ gives $A_r/A_i=-k_i/k_r$ in that more complete boundary relation. Thus the printed equal-amplitude assertion requires the nondegenerate leading <quasi-geostrophic approximation>, or an explicit homogeneous wall condition; it is not a general exact shallow-water reflection law.
Back to article page
