1 Introduction
Non-divergent waves in a two-dimensional fluid layer on a rotating sphere, restored solely by Coriolis forces due to vorticity conservation, are present as second-class waves in Laplace’s equations and were studied by Margules (Reference Margules1893) and Hough (Reference Hough1898). Their nature was clarified when Rossby & Collaborators (Reference Rossby1939) derived a simplified version of the barotropic vorticity equation on a mid-latitude tangent plane (
$\unicode[STIX]{x1D6FD}$
-plane). They eliminated the first-class gravity waves by assuming a rigid lid. Thus they considered non-divergent flow in a fluid layer in which vorticity changes in time due to advection of variations in background planetary vorticity. Linear, free-wave solutions of this equation display strictly westward phase propagation, but west or eastward energy propagation for long and short waves, respectively. In recognition of the value of this clarification, these waves are today called Rossby waves. They are forced by the curl of the wind stress and the barotropic non-divergent linear motions are governed by
$$\begin{eqnarray}\unicode[STIX]{x2202}_{t}q+\unicode[STIX]{x1D6FD}v=\text{curl}\,\unicode[STIX]{x1D749}.\end{eqnarray}$$
The relative vorticity
$q=\unicode[STIX]{x2202}_{x}v-\unicode[STIX]{x2202}_{y}u$
is created by south–north advection of planetary vorticity and by
$\unicode[STIX]{x1D749}=\{\unicode[STIX]{x1D70F}_{x},\unicode[STIX]{x1D70F}_{y}\}$
the wind-stress vector (see e.g. Veronis Reference Veronis1958; Pedlosky Reference Pedlosky1987). As usual,
$t$
is time,
$x$
longitude,
$y$
latitude and
$\unicode[STIX]{x1D6FD}v$
is the advection of planetary vorticity by the latitudinal (south–north) velocity component
$v$
.
Introducing a streamfunction
$\unicode[STIX]{x1D713}$
, so that the velocity components are
$u=-\unicode[STIX]{x2202}_{y}\unicode[STIX]{x1D713},v=\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D713}$
and
$q=\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D713}$
, equation (1.1) becomes
$\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D713}+\unicode[STIX]{x1D6FD}\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D713}=\text{curl}\,\unicode[STIX]{x1D749}$
with
$\unicode[STIX]{x1D6FB}^{2}=(\unicode[STIX]{x2202}_{x}^{2}+\unicode[STIX]{x2202}_{y}^{2})$
. In this study, the Green’s function
$G_{\unicode[STIX]{x1D6FF}}$
for an ‘impulsive point source’ and
$G_{H}$
for a ‘switch-on point source’ are defined as solutions of
$$\begin{eqnarray}{\mathcal{L}}G_{\unicode[STIX]{x1D6FF}}=\unicode[STIX]{x1D6FF}(t)\unicode[STIX]{x1D6FF}(x)\unicode[STIX]{x1D6FF}(y),\quad {\mathcal{L}}G_{H}=H(t)\unicode[STIX]{x1D6FF}(x)\unicode[STIX]{x1D6FF}(y),\quad {\mathcal{L}}=\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D6FB}^{2}+\unicode[STIX]{x1D6FD}\unicode[STIX]{x2202}_{x}.\end{eqnarray}$$
${\mathcal{L}}$
is the Rossby wave operator,
$\unicode[STIX]{x1D6FF}$
the usual Dirac delta function and
$H$
the Heaviside unit step function and by definition
$\unicode[STIX]{x1D6FF}(t)\equiv \text{d}H(t)/\text{d}t$
. In this paper we shall show that
$$\begin{eqnarray}\displaystyle G_{\unicode[STIX]{x1D6FF}} & = & \displaystyle {\displaystyle \frac{H(t)}{4}}[\text{J}_{0}(z_{+})\text{Y}_{0}(z_{-})+\text{J}_{0}(z_{-})\text{Y}_{0}(z_{+})],\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle G_{H} & = & \displaystyle {\displaystyle \frac{tH(t)}{2(z_{+}^{2}-z_{-}^{2})}} [\!z_{+}\text{J}_{1}(z_{+})\text{Y}_{0}(z_{-})-z_{-}\text{J}_{1}(z_{-})\text{Y}_{0}(z_{+})\nonumber\\ \displaystyle & & \displaystyle +\,z_{+}\text{J}_{0}(z_{-})\text{Y}_{1}(z_{+})-z_{-}\text{J}_{0}(z_{+})\text{Y}_{1}(z_{-})\!].\end{eqnarray}$$
In (1.3) and (1.4)
$\text{J}_{n},\text{Y}_{n}$
are
$n$
th-order Bessel functions of the first and second kind, respectively, with complex conjugate arguments
$$\begin{eqnarray}z_{+}=\sqrt{\unicode[STIX]{x1D6FD}tr}\text{e}^{+\text{i}\unicode[STIX]{x1D703}/2},\quad z_{-}=\sqrt{\unicode[STIX]{x1D6FD}tr}\text{e}^{-\text{i}\unicode[STIX]{x1D703}/2}\quad \text{or}\quad z_{\pm }=\sqrt{\unicode[STIX]{x1D6FD}t(x\pm \text{i}y)}.\end{eqnarray}$$
The usual cylindrical coordinate system
$\{r,\unicode[STIX]{x1D703}\}$
is employed with the origin at the forcing location (see figure 1). The prefactor
$H(t)$
in (1.3) and (1.4) enforces ‘causality’, i.e.
$G_{\unicode[STIX]{x1D6FF},H}=0$
for
$t<0$
. Since efficient algorithms for Bessel functions exist, the compact expressions in (1.3) and (1.4) allow for a quick evaluation: with a few lines of code in little time a movie can be created visualizing the temporal and spatial evolutions.
The Cartesian
$\{x,y\}$
and cylindrical
$\{r,\unicode[STIX]{x1D703}\}$
coordinate systems. The forcing location for
$G_{\unicode[STIX]{x1D6FF}}$
and
$G_{H}$
at
$r=0$
is indicated by ●. ‘East’ is defined by
$\unicode[STIX]{x1D703}=0$
, ‘west’ by
$\unicode[STIX]{x1D703}=\pm \unicode[STIX]{x03C0}$
, ‘north’ by
$\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2$
,
$r$
is the distance from the source and
$x=r\cos \unicode[STIX]{x1D703},y=r\sin \unicode[STIX]{x1D703}$
.
Typically in finding Green’s functions and solutions of linear partial differential equations like (1.2), the well-established theory for Laplace and/or Fourier transforms is used in combination with the theory of differential equations. Various integral representations for
$G_{\unicode[STIX]{x1D6FF}}$
and
$G_{H}$
that have been derived in such a manner are briefly reviewed in § 2. They also reveal a relation between
$G_{\unicode[STIX]{x1D6FF}}$
and
$G_{H}$
: not only is
$\unicode[STIX]{x2202}_{t}G_{H}=G_{\unicode[STIX]{x1D6FF}}$
, as expected, but once
$G_{\unicode[STIX]{x1D6FF}}$
is known
$G_{H}$
follows by differentiation of
$G_{\unicode[STIX]{x1D6FF}}$
with respect to the polar angle
$\unicode[STIX]{x1D703}$
. Precise numerical evaluation of the integral representation is cumbersome, but for locations exactly east
$(\unicode[STIX]{x1D703}=0)$
and west (
$\unicode[STIX]{x1D703}=\pm \unicode[STIX]{x03C0}$
) of the source they can be evaluated exactly. Our new expressions for
$G_{\unicode[STIX]{x1D6FF}}$
and
$G_{H}$
provide a complete ‘picture’ at any angle
$\unicode[STIX]{x1D703}$
with the east–west axis.
The compact formula (1.3) for
$G_{\unicode[STIX]{x1D6FF}}$
was discovered by recognizing that a particular integral representation of
$G_{\unicode[STIX]{x1D6FF}}$
is essentially an integral of the kind studied by Dixon & Ferrar (Reference Dixon and Ferrar1933). Differentiation of (1.3) then quickly yielded
$G_{H}$
given by (1.4). This is discussed in § 3 where, for completeness’ sake, we also show that the Green’s functions given by (1.3) and (1.4) indeed solve (1.2). In § 4 we provide illustrative graphs of
$G_{\unicode[STIX]{x1D6FF}}$
and
$G_{H}$
and derive simple approximations which presume large arguments
$|z_{\pm }|$
or
$\sqrt{\unicode[STIX]{x1D6FD}tr}\gg 1$
. Also in § 4 the associated wave–energy density distributions are considered. In § 5 we discuss the ramifications of this study and in particular we draw attention to results that indicate that parabolic coordinates
$\unicode[STIX]{x1D701}=\sqrt{r+x},\,\unicode[STIX]{x1D702}=\sqrt{r-x}$
appear to be a natural choice when studying non-divergent Rossby waves.
2 Integral representations and relations between
$G_{\unicode[STIX]{x1D6FF}}$
and
$G_{H}$
Veronis (Reference Veronis1958) showed that
$$\begin{eqnarray}G_{\unicode[STIX]{x1D6FF}}=-{\displaystyle \frac{H(t)}{\unicode[STIX]{x03C0}}}\int _{0}^{\infty }{\displaystyle \frac{\text{J}_{0}\left(2\sqrt{\unicode[STIX]{x1D6FD}tr\left(\unicode[STIX]{x1D702}^{2}+\cos ^{2}{\textstyle \frac{1}{2}}\unicode[STIX]{x1D703}\right)}\right)}{\sqrt{\unicode[STIX]{x1D702}^{2}+1}}}\,\text{d}\unicode[STIX]{x1D702},\end{eqnarray}$$
correcting for a typographical error. He called the point-size burst of wind-stress curl a ‘tweak’ and derived (2.1) through the use of a Laplace transform in time of (1.2a
) which reduced the problem to solving a second-order partial differential equation (in variables
$x,y$
). With the solution at hand, equation (2.1) followed from the inverse Laplace transform. Longuet-Higgins (Reference Longuet-Higgins1965) also found (2.1), but through a triple Fourier transform. Much later, Kamenkovich (Reference Kamenkovich1989) established that
$$\begin{eqnarray}G_{H}=-{\displaystyle \frac{H(t)}{\unicode[STIX]{x03C0}}}\sqrt{{\displaystyle \frac{t}{\unicode[STIX]{x1D6FD}r}}}\int _{0}^{\infty }{\displaystyle \frac{\text{J}_{1}\left(2\sqrt{\unicode[STIX]{x1D6FD}tr\left(\unicode[STIX]{x1D702}^{2}+\cos ^{2}{\textstyle \frac{1}{2}}\unicode[STIX]{x1D703}\right)}\right)}{\sqrt{\unicode[STIX]{x1D702}^{2}+1}\sqrt{\unicode[STIX]{x1D702}^{2}+\cos ^{2}{\textstyle \frac{1}{2}}\unicode[STIX]{x1D703}}}}\,\text{d}\unicode[STIX]{x1D702},\end{eqnarray}$$
using a Laplace transform of (1.2b
) with respect to time. Both infinite integrals defining
$G_{\unicode[STIX]{x1D6FF},H}$
are difficult to evaluate numerically due to the oscillatory behaviour of
$\text{J}_{0}$
,
$\text{J}_{1}$
for increasing
$\unicode[STIX]{x1D702}$
. However, computationally more efficient integral expressions for
$G_{\unicode[STIX]{x1D6FF}}$
have been found by Llewellyn-Smith (Reference Llewellyn-Smith1997).
The integral representation (2.1) for
$G_{\unicode[STIX]{x1D6FF}}$
reduces on the east–west axis to
$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D703}=0\text{ (east)}:\quad G_{\unicode[STIX]{x1D6FF}}={\displaystyle \frac{\text{J}_{0}(\sqrt{\unicode[STIX]{x1D6FD}tr})\text{Y}_{0}(\sqrt{\unicode[STIX]{x1D6FD}tr})}{2}}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D703}=\pm \unicode[STIX]{x03C0}\text{(west)}:\quad G_{\unicode[STIX]{x1D6FF}}=-{\displaystyle \frac{\text{I}_{0}(\sqrt{\unicode[STIX]{x1D6FD}tr})\text{K}_{0}(\sqrt{\unicode[STIX]{x1D6FD}tr})}{\unicode[STIX]{x03C0}}}, & \displaystyle\end{eqnarray}$$
$\text{I}_{n}$
,
$\text{K}_{n}$
the
$n$
th-order modified Bessel functions of the first and second kind, respectively. For convenience we have dropped the ‘causality switch’
$H(t)$
. The limiting case
$\unicode[STIX]{x1D703}=\pm \unicode[STIX]{x03C0}$
(2.3b
) was first noted, apart from a missing minus sign, by Longuet-Higgins (Reference Longuet-Higgins1965) but for
$\unicode[STIX]{x1D703}=0$
he erred. The correct limit
$\unicode[STIX]{x1D703}=0$
in (2.3a
) was given by Llewellyn-Smith (Reference Llewellyn-Smith1997). Kamenkovich (Reference Kamenkovich1989) recognized with the integral representation (2.2) that
$G_{H}$
reduces on the east–west axis to 
$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D703}=0\text{ (east)}:\quad G_{H}={\displaystyle \frac{t}{2}}[\text{J}_{0}(\sqrt{\unicode[STIX]{x1D6FD}tr})\text{Y}_{0}(\sqrt{\unicode[STIX]{x1D6FD}tr})+\text{J}_{1}(\sqrt{\unicode[STIX]{x1D6FD}tr})\text{Y}_{1}(\sqrt{\unicode[STIX]{x1D6FD}tr})], & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D703}=\pm \unicode[STIX]{x03C0}\text{(west)}:\quad G_{H}=-{\displaystyle \frac{t}{\unicode[STIX]{x03C0}}}[\text{I}_{0}(\sqrt{\unicode[STIX]{x1D6FD}tr})\text{K}_{0}(\sqrt{\unicode[STIX]{x1D6FD}tr})+\text{I}_{1}(\sqrt{\unicode[STIX]{x1D6FD}tr})\text{K}_{1}(\sqrt{\unicode[STIX]{x1D6FD}tr})]. & \displaystyle\end{eqnarray}$$
Simpler integral expressions
$$\begin{eqnarray}\displaystyle & \displaystyle G_{\unicode[STIX]{x1D6FF}}=-{\displaystyle \frac{1}{4\unicode[STIX]{x03C0}}}\int _{-\infty }^{+\infty }\text{J}_{0}(\sqrt{2\unicode[STIX]{x1D6FD}tr(\cosh \unicode[STIX]{x1D708}+\cos \unicode[STIX]{x1D703})})\,\text{d}\unicode[STIX]{x1D708}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle G_{H}=-{\displaystyle \frac{t}{2\unicode[STIX]{x03C0}}}\int _{-\infty }^{+\infty }{\displaystyle \frac{\text{J}_{1}(\sqrt{2\unicode[STIX]{x1D6FD}tr(\cosh \unicode[STIX]{x1D708}+\cos \unicode[STIX]{x1D703})})}{\sqrt{2\unicode[STIX]{x1D6FD}tr(\cosh \unicode[STIX]{x1D708}+\cos \unicode[STIX]{x1D703})}}}\text{d}\unicode[STIX]{x1D708}, & \displaystyle\end{eqnarray}$$
$\unicode[STIX]{x1D702}=\sinh 1/2\unicode[STIX]{x1D708}$
. Note that
$G_{\unicode[STIX]{x1D6FF}}$
is a function of the similarity variables
$\unicode[STIX]{x1D6FD}tx,\,\unicode[STIX]{x1D6FD}ty$
or
$G_{\unicode[STIX]{x1D6FF}}=F(\unicode[STIX]{x1D6FD}tr,\unicode[STIX]{x1D703})$
while
$G_{H}$
is of the form
$tF(\unicode[STIX]{x1D6FD}tr,\unicode[STIX]{x1D703})$
.
To proceed, observe how on the east side (
$\unicode[STIX]{x1D703}=0$
), the integral expression (2.5a
) for
$G_{\unicode[STIX]{x1D6FF}}$
simplifies to
$$\begin{eqnarray}\displaystyle G_{\unicode[STIX]{x1D6FF}} & = & \displaystyle -{\displaystyle \frac{1}{4\unicode[STIX]{x03C0}}}\int _{-\infty }^{+\infty }\!\text{J}_{0}\left(2\sqrt{\unicode[STIX]{x1D6FD}tr}\,\cosh \frac{1}{2}\unicode[STIX]{x1D708}\right)\,\text{d}\unicode[STIX]{x1D708}\nonumber\\ \displaystyle & = & \displaystyle -{\displaystyle \frac{1}{2\unicode[STIX]{x03C0}^{2}}}\iint _{-\infty }^{+\infty }\!\sin (2\sqrt{\unicode[STIX]{x1D6FD}tr}\,\cosh \unicode[STIX]{x1D708}^{\prime }\cosh \unicode[STIX]{x1D707})\,\text{d}\unicode[STIX]{x1D708}^{\prime }\,\text{d}\unicode[STIX]{x1D707},\end{eqnarray}$$
where we use
$\unicode[STIX]{x1D708}^{\prime }=(1/2)\unicode[STIX]{x1D708}$
, and the first of the integral expressions for the Bessel functions
$$\begin{eqnarray}\text{J}_{0}(x)={\displaystyle \frac{1}{\unicode[STIX]{x03C0}}}\int _{-\infty }^{+\infty }\sin (x\,\cosh \unicode[STIX]{x1D707})\,\text{d}\unicode[STIX]{x1D707},\quad \text{Y}_{0}(x)=-{\displaystyle \frac{1}{\unicode[STIX]{x03C0}}}\int _{-\infty }^{+\infty }\cos (x\,\cosh \unicode[STIX]{x1D707})\,\text{d}\unicode[STIX]{x1D707}.\end{eqnarray}$$
Setting
$m=\unicode[STIX]{x1D707}+\unicode[STIX]{x1D708}^{\prime },\;n=\unicode[STIX]{x1D707}-\unicode[STIX]{x1D708}^{\prime }$
, we rotate integration variables
$\unicode[STIX]{x1D707},\;\unicode[STIX]{x1D708}^{\prime }$
over
$\unicode[STIX]{x03C0}/4$
, yielding
$$\begin{eqnarray}G_{\unicode[STIX]{x1D6FF}}=-{\displaystyle \frac{1}{4\unicode[STIX]{x03C0}^{2}}}\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }\sin (\sqrt{\unicode[STIX]{x1D6FD}tr}[\cosh m+\cosh n])\,\text{d}m\,\text{d}n.\end{eqnarray}$$
Using the trigonometric identity
$\sin (a+b)=\sin a\cos b+\sin b\cos a$
, this integral is separable and leads, with the Bessel identities (2.7a,b
), to (2.3a
), i.e.
$$\begin{eqnarray}\displaystyle G_{\unicode[STIX]{x1D6FF}} & = & \displaystyle -{\displaystyle \frac{1}{4\unicode[STIX]{x03C0}^{2}}}\left(\int _{-\infty }^{+\infty }\sin (\sqrt{\unicode[STIX]{x1D6FD}tr}\,\cosh m)\,\text{d}m\int _{-\infty }^{+\infty }\cos (\sqrt{\unicode[STIX]{x1D6FD}tr}\,\cosh n)\text{d}n\right.\nonumber\\ \displaystyle & & \displaystyle +\,\left.\int _{-\infty }^{+\infty }\cos (\sqrt{\unicode[STIX]{x1D6FD}tr}\,\cosh m)\,\text{d}m\int _{-\infty }^{+\infty }\sin (\sqrt{\unicode[STIX]{x1D6FD}tr}\,\cosh n)\,\text{d}n\right)={\displaystyle \frac{\text{J}_{0}(\unicode[STIX]{x1D70C})\text{Y}_{0}(\unicode[STIX]{x1D70C})}{2}},\qquad\end{eqnarray}$$
with
$\unicode[STIX]{x1D70C}\equiv \sqrt{\unicode[STIX]{x1D6FD}tr}$
. A similar derivation can be used to obtain the established result (2.3b
) on the west axis (
$\unicode[STIX]{x1D703}=\pm \unicode[STIX]{x03C0}$
).
Since
$G_{H}$
describes the response to sustained forcing represented by the Heaviside function
$H(t)$
and
$G_{\unicode[STIX]{x1D6FF}}$
to impulsive forcing represented by
$\unicode[STIX]{x1D6FF}(t)$
, one expects
$G_{\unicode[STIX]{x1D6FF}}=\unicode[STIX]{x2202}G_{H}/\unicode[STIX]{x2202}t$
. The integral representations for
$G_{\unicode[STIX]{x1D6FF}}$
and
$G_{H}$
confirm this. For example, in (2.5b
)
$$\begin{eqnarray}G_{H}={\displaystyle \frac{1}{2\unicode[STIX]{x03C0}}}\int _{-\infty }^{+\infty }{\displaystyle \frac{t\text{J}_{0}^{\prime }(z)}{z}}\,\text{d}\unicode[STIX]{x1D708},\quad z\equiv \sqrt{2\unicode[STIX]{x1D6FD}tr(\cosh \unicode[STIX]{x1D708}+\cos \unicode[STIX]{x1D703})},\end{eqnarray}$$
where we used the fact that
$\text{J}_{1}=-\text{J}_{0}^{\prime }$
(a prime indicates differentiation). Since
$\unicode[STIX]{x2202}z/\unicode[STIX]{x2202}t=z/2t$
, it follows that
$$\begin{eqnarray}\displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}G_{H}}{\unicode[STIX]{x2202}t}} & = & \displaystyle {\displaystyle \frac{1}{2\unicode[STIX]{x03C0}}}\left[\int {\displaystyle \frac{\text{J}_{0}^{\prime }(z)}{z}}\text{d}\unicode[STIX]{x1D708}+\int t\left({\displaystyle \frac{\text{J}_{0}^{\prime \prime }(z)}{z}}{\displaystyle \frac{z}{2t}}-{\displaystyle \frac{\text{J}_{0}^{\prime }(z)}{z^{2}}}{\displaystyle \frac{z}{2t}}\right)\,\text{d}\unicode[STIX]{x1D708}\right]\nonumber\\ \displaystyle & = & \displaystyle {\displaystyle \frac{1}{2\unicode[STIX]{x03C0}}}\left[\int {\displaystyle \frac{1}{2}}\left({\displaystyle \frac{\text{J}_{0}^{\prime }(z)}{z}}+\text{J}_{0}^{\prime \prime }(z)\right)\,\text{d}\unicode[STIX]{x1D708}\right]=-{\displaystyle \frac{1}{4\unicode[STIX]{x03C0}}}\int \text{J}_{0}(z)\,\text{d}\unicode[STIX]{x1D708}=(2.5a)=G_{\unicode[STIX]{x1D6FF}}\qquad\end{eqnarray}$$
because
$\text{J}_{0}$
solves the Bessel equation
$z^{2}\text{J}_{0}^{\prime \prime }+z\text{J}_{0}^{\prime }+z^{2}\text{J}_{0}=0$
.
Another relation exists between
$G_{\unicode[STIX]{x1D6FF}}$
and
$G_{H}$
: in (2.5b
)
$$\begin{eqnarray}{\displaystyle \frac{t\text{J}_{1}(z)}{z}}={\displaystyle \frac{1}{\unicode[STIX]{x1D6FD}r\sin \unicode[STIX]{x1D703}}}{\displaystyle \frac{\unicode[STIX]{x2202}\text{J}_{0}(z)}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}}.\end{eqnarray}$$
Comparison with (2.5a ) shows that
$$\begin{eqnarray}G_{H}={\displaystyle \frac{2}{\unicode[STIX]{x1D6FD}r\sin \unicode[STIX]{x1D703}}}{\displaystyle \frac{\unicode[STIX]{x2202}\,G_{\unicode[STIX]{x1D6FF}}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}}\quad \text{and}\quad \left({\displaystyle \frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}t\unicode[STIX]{x2202}\unicode[STIX]{x1D703}}}-{\displaystyle \frac{1}{2}}\unicode[STIX]{x1D6FD}r\sin \unicode[STIX]{x1D703}\right)G_{H}=0\end{eqnarray}$$
because
$G_{\unicode[STIX]{x1D6FF}}=\unicode[STIX]{x2202}_{t}G_{H}$
. Time differentiation shows that therefore
$G_{\unicode[STIX]{x1D6FF}}$
also satisfies (2.13b
).
3 Closed-form expressions for
$G_{\unicode[STIX]{x1D6FF}}$
and
$G_{H}$
In view of the definition of
$z_{\pm }$
in (1.5), the integral representation (2.5a
) equals
$$\begin{eqnarray}G_{\unicode[STIX]{x1D6FF}}=-{\displaystyle \frac{1}{4\unicode[STIX]{x03C0}}}\int _{-\infty }^{+\infty }\text{J}_{0}\left(\sqrt{z_{+}^{2}+z_{-}^{2}+2z_{+}z_{-}\,\cosh \unicode[STIX]{x1D708}}\right)\,\text{d}\unicode[STIX]{x1D708}.\end{eqnarray}$$
This observation led us to the expression (1.3) for
$G_{\unicode[STIX]{x1D6FF}}$
as follows. Dixon & Ferrar (Reference Dixon and Ferrar1933) considered products of modified Bessel functions of the second kind
$\text{K}_{\unicode[STIX]{x1D708},\unicode[STIX]{x1D707}}$
and showed that such products can be reduced to a single integral involving
$\text{K}_{\unicode[STIX]{x1D708}+\unicode[STIX]{x1D707}}$
, provided certain conditions are met. For the purpose of this paper it suffices to point out that one of their results implies that for real
$x,y>0$
$$\begin{eqnarray}\text{K}_{0}(\text{i}x)\text{K}_{0}(\text{i}y)=\int _{-\infty }^{+\infty }\text{K}_{0}(\text{i}\unicode[STIX]{x1D706})\,\text{d}t^{\prime },\quad \unicode[STIX]{x1D706}=\sqrt{x^{2}+y^{2}+2xy\cosh 2t^{\prime }}\end{eqnarray}$$
(Dixon & Ferrar Reference Dixon and Ferrar1933, formula 3.32). Connection formulas tell us further that
$$\begin{eqnarray}\text{K}_{0}(z)=-{\textstyle \frac{1}{2}}\unicode[STIX]{x03C0}\text{i}H_{0}^{(2)}\left(-\text{i}z\right),\quad -{\textstyle \frac{1}{2}}\unicode[STIX]{x03C0}\leqslant \operatorname{ph}z\leqslant \unicode[STIX]{x03C0},\end{eqnarray}$$
with the Hankel function
$H_{0}^{(2)}=\text{J}_{0}-\text{i}\text{Y}_{0}$
(see the NIST Handbook of Mathematical Functions, Olver et al. (Reference Olver, Lozier, Boisvert and Clark2010), formula 10.27.8). Specifically this means
$\text{K}_{0}(\text{i}x)=-(1/2)\unicode[STIX]{x03C0}\text{i}H_{0}^{(2)}(x)$
and likewise for
$y$
. Therefore with (3.2) it follows that
$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{-\infty }^{+\infty }[\text{J}_{0}(\unicode[STIX]{x1D706})-\text{i}\text{Y}_{0}(\unicode[STIX]{x1D706})]\,\text{d}t^{\prime }=\int _{-\infty }^{+\infty }H_{0}^{(2)}(\unicode[STIX]{x1D706})\,\text{d}t^{\prime }=-{\displaystyle \frac{\unicode[STIX]{x03C0}\text{i}}{2}}H_{0}^{(2)}(x)H_{0}^{(2)}(y)\nonumber\\ \displaystyle & & \displaystyle \quad =-{\displaystyle \frac{\unicode[STIX]{x03C0}}{2}}[\text{J}_{0}(x)\text{Y}_{0}(y)+\text{J}_{0}(y)\text{Y}_{0}(x)]-{\displaystyle \frac{\unicode[STIX]{x03C0}\text{i}}{2}}[\text{J}_{0}(x)\text{J}_{0}(y)-\text{Y}_{0}(x)\text{Y}_{0}(y)]\end{eqnarray}$$
(for the more general case involving
$H_{\unicode[STIX]{x1D708}}^{(2)}(x)H_{\unicode[STIX]{x1D707}}^{(2)}(y)$
see formula 17.4.2 (76) in Erdélyi (Reference Erdélyi1953) and for related results Magnus, Oberhettinger & Soni (Reference Magnus, Oberhettinger and Soni1966), p. 93, which however contains a typographical error). Equating real parts in (3.4) leads to the product
$\text{J}_{0}(x)\text{Y}_{0}(y)+\text{J}_{0}(y)\text{Y}_{0}(x)$
expressed as an integral over
$\text{J}_{0}(\unicode[STIX]{x1D706})$
. Comparison with the integral expression (3.1) shows that the closed-form expression (1.3) would follow if we could simply substitute
$x=z_{+},\,y=z_{-}$
in (3.4) after setting
$\unicode[STIX]{x1D708}=2t^{\prime }$
in (3.1). Thus we surmised that
$G_{\unicode[STIX]{x1D6FF}}$
given in (1.3) is the solution of (1.2a
). This ‘educated guess’ proved to be correct by direct verification (see below).
However, Llewellyn Smith (University of California, San Diego) pointed out during the revision process of this manuscript that the results of Dixon & Ferrar (Reference Dixon and Ferrar1933) in fact imply that (3.1) equals (1.3) when the principle of analytic continuation is invoked. Again tailored specifically to our needs here, Dixon & Ferrar (Reference Dixon and Ferrar1933) state that for complex variables
$Z,z$
with positive real parts (absolute phase
${<}\unicode[STIX]{x03C0}/2$
)
$$\begin{eqnarray}\text{K}_{0}(Z)\text{K}_{0}(z)=\int _{-\infty }^{+\infty }\text{K}_{0}\left(\sqrt{Z^{2}+z^{2}+2Zz\,\cosh 2t^{\prime }}\right)\text{d}t^{\prime },\quad |\text{ph}\,Z,\,z|<{\displaystyle \frac{\unicode[STIX]{x03C0}}{2}}.\end{eqnarray}$$
The Bessel functions are analytic when a branch cut is made on the negative real axis, i.e. for
$\mathbb{C}\backslash (-\infty ,0]$
. If we exclude the west axis (
$\unicode[STIX]{x1D703}=\pm \unicode[STIX]{x03C0}$
), equation (3.5) is valid when
$Z=z_{+},z=z_{-}$
is substituted (for
$\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}$
we have
$z_{\pm }=\pm \text{i}\sqrt{\unicode[STIX]{x1D6FD}tr}$
and for
$\unicode[STIX]{x1D703}=-\unicode[STIX]{x03C0}$
we have
$z_{\pm }=\mp \text{i}\sqrt{\unicode[STIX]{x1D6FD}tr}$
according to (1.5)). Both variables lie in the open right half of the complex plane and if rotated anti-clockwise by an angle
$\unicode[STIX]{x1D719}$
they become
$z_{\pm }^{\prime }=\text{e}^{\text{i}\unicode[STIX]{x1D719}}z_{\pm }$
. For
$\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}/2$
the rotated
$z_{\pm }^{\prime }=\text{i}z_{\pm }$
have been moved to the upper half of the complex plane without crossing the negative real axis. Therefore the left-hand side of (3.5) remains analytic. The argument of
$\text{K}_{0}$
in the integral becomes under this rotation simply
$\text{e}^{\text{i}\unicode[STIX]{x1D719}}\sqrt{2\unicode[STIX]{x1D6FD}tr(\cos \unicode[STIX]{x1D703}+\cosh 2t^{\prime })}$
. Thus, if the west axis is excluded from consideration, as
$\unicode[STIX]{x1D719}$
is increased towards
$\unicode[STIX]{x03C0}/2$
the argument never crosses the branch cut of
$\text{K}_{0}$
and the integrand and the integral on the right-hand side of (3.5) remain analytic. By analytic continuation we therefore find that
$$\begin{eqnarray}\text{K}_{0}(\text{i}z_{+})\text{K}_{0}(\text{i}z_{-})=\int _{-\infty }^{+\infty }\text{K}_{0}\left(\text{i}\sqrt{z_{+}^{2}+z_{-}^{2}+2z_{+}z_{-}\,\cosh 2t^{\prime }}\right)\,\text{d}t^{\prime }.\end{eqnarray}$$
The conditions for (3.3) to be valid are met and we can, just as shown above for real
$x,y$
, equate the real parts on the left and right in (3.6). This establishes that the integral representation (3.1) equals the closed-form expression (1.3) for
$G_{\unicode[STIX]{x1D6FF}}$
. Despite this compelling argument, we nonetheless verify the validity of (1.3) below and show how
$G_{H}$
(1.4) was derived.
The parabolic coordinate system
$\{\unicode[STIX]{x1D701},\unicode[STIX]{x1D702}\}$
with
$\unicode[STIX]{x1D701}=\sqrt{r+x},\,\unicode[STIX]{x1D702}=\sqrt{r-x}$
defined in (3.8). The entire western axis
$x<0,\,y=0$
coincides with
$\unicode[STIX]{x1D701}=0$
, the eastern axis
$x>0,\,y=0$
with
$\unicode[STIX]{x1D702}=0$
. The origin
$\unicode[STIX]{x1D701}=\unicode[STIX]{x1D702}=0$
indicated by ● is the forcing location as in figure 1.
3.1 Validation of
$G_{\unicode[STIX]{x1D6FF}}$
In order to verify that
$G_{\unicode[STIX]{x1D6FF}}$
(1.3) is correct, it is expedient to write the complex conjugate variables
$z_{\pm }$
introduced in (1.5) as
$$\begin{eqnarray}z_{+}=az_{\star },\quad z_{-}=a\bar{z}_{\star },\quad a=\sqrt{{\displaystyle \frac{\unicode[STIX]{x1D6FD}t}{2}}},\quad z_{\star }\equiv \unicode[STIX]{x1D701}+\text{i}\unicode[STIX]{x1D702},\quad \bar{z}_{\star }\equiv \unicode[STIX]{x1D701}-\text{i}\unicode[STIX]{x1D702},\end{eqnarray}$$
with parabolic coordinates
$$\begin{eqnarray}\unicode[STIX]{x1D701}=\sqrt{r(1+\cos \unicode[STIX]{x1D703})}=\sqrt{r+x},\quad \unicode[STIX]{x1D702}=\sqrt{r(1-\cos \unicode[STIX]{x1D703})}=\sqrt{r-x}.\end{eqnarray}$$
Lines of constant
$\unicode[STIX]{x1D701}$
are parabolas that open towards the west (
$x<0$
) and constant
$\unicode[STIX]{x1D702}$
parabolas that open towards the east (
$x>0$
) as sketched in figure 2. The Laplace operator in these coordinates is well known (see Moon & Spencer Reference Moon and Spencer1961) while
$\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x$
is easily calculated with the chain rule. They are
$$\begin{eqnarray}\unicode[STIX]{x1D6FB}^{2}={\displaystyle \frac{1}{\unicode[STIX]{x1D701}^{2}+\unicode[STIX]{x1D702}^{2}}}\left[{\displaystyle \frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}^{2}}}+{\displaystyle \frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{2}}}\right],\quad {\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}}={\displaystyle \frac{1}{2r}}\left[\unicode[STIX]{x1D701}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}}-\unicode[STIX]{x1D702}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}}\right].\end{eqnarray}$$
But
$2r=\unicode[STIX]{x1D701}^{2}+\unicode[STIX]{x1D702}^{2}$
and the Rossby wave operator becomes
$$\begin{eqnarray}{\mathcal{L}}={\displaystyle \frac{1}{\unicode[STIX]{x1D701}^{2}+\unicode[STIX]{x1D702}^{2}}}\left\{{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}}\left({\displaystyle \frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}^{2}}}+{\displaystyle \frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}^{2}}}\right)+\unicode[STIX]{x1D6FD}\left(\unicode[STIX]{x1D701}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}}-\unicode[STIX]{x1D702}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}}\right)\right\}.\end{eqnarray}$$
With a few elementary manipulations one finds next that
$$\begin{eqnarray}{\mathcal{L}}={\displaystyle \frac{1}{z_{\star }\bar{z}_{\star }}}\left\{4{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}}\left({\displaystyle \frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}z_{\star }\unicode[STIX]{x2202}\bar{z}_{\star }}}\right)+\unicode[STIX]{x1D6FD}\left(\bar{z}_{\star }{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z_{\star }}}+z_{\star }{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\bar{z}_{\star }}}\right)\right\}.\end{eqnarray}$$
With (3.11) it follows, after collecting terms, that
$$\begin{eqnarray}\displaystyle {\mathcal{L}}\,\text{J}_{0}(z_{+})\text{Y}_{0}(z_{-}) & = & \displaystyle {\displaystyle \frac{\unicode[STIX]{x1D6FD}}{z_{\star }\bar{z}_{\star }}}\left\{\text{J}_{0}^{\prime }(z_{+})[z_{-}\text{Y}_{0}^{\prime \prime }(z_{-})+\text{Y}_{0}^{\prime }(z_{-})+z_{-}\text{Y}_{0}(z_{-})]\right.\nonumber\\ \displaystyle & & \displaystyle \left.+\,\text{Y}_{0}^{\prime }(z_{-})[z_{+}\text{J}_{0}^{\prime \prime }(z_{+})+\text{J}_{0}^{\prime }(z_{+})+z_{+}\text{J}_{0}(z_{+})]\right\}=0\end{eqnarray}$$
because
$\text{J}_{0},\text{Y}_{0}$
satisfy the Bessel equation
$$\begin{eqnarray}z^{2}{\mathcal{J}}_{0}^{\prime \prime }+z{\mathcal{J}}_{0}^{\prime }+z^{2}{\mathcal{J}}_{0}=0.\end{eqnarray}$$
Clearly also
${\mathcal{L}}\text{J}_{0}(z_{-})\text{Y}_{0}(z_{+})=0$
and away from
$z_{\star }\bar{z}_{\star }=2r=0$
for
$t>0$
indeed
${\mathcal{L}}G_{\unicode[STIX]{x1D6FF}}=0$
.
The
$\unicode[STIX]{x1D6FF}(t)\unicode[STIX]{x1D6FF}(x)\unicode[STIX]{x1D6FF}(y)$
singularity in (1.2a
) has to come from the term
$\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D6FB}^{2}G_{\unicode[STIX]{x1D6FF}}$
. The
$\unicode[STIX]{x1D6FF}(t)$
behaviour must be associated with the time derivative of
$H(t)$
which multiplies
$\text{J}_{0}(z_{+})\text{Y}_{0}(z_{-})+\text{J}_{0}(z_{-})\text{Y}_{0}(z_{+})$
in (1.3). Replacing
$\unicode[STIX]{x1D6FF}(x)\unicode[STIX]{x1D6FF}(y)$
on the right-hand side of (1.2a
) by
$\unicode[STIX]{x1D6FF}(r)/(2\unicode[STIX]{x03C0}r)$
, for small
$r$
the Green’s function must therefore behave as
$G_{\unicode[STIX]{x1D6FF}}\approx H(t)\ln r/2\unicode[STIX]{x03C0}$
. That this is true is seen as follows: since for small
$z$
$$\begin{eqnarray}z\rightarrow 0:\quad \text{J}_{0}(z)\approx 1,\quad \text{Y}_{0}(z)\approx (2/\unicode[STIX]{x03C0})\ln z,\end{eqnarray}$$
near the origin
$$\begin{eqnarray}r\downarrow 0:~G_{\unicode[STIX]{x1D6FF}}\approx {\displaystyle \frac{H(t)}{4}}\left[{\displaystyle \frac{2}{\unicode[STIX]{x03C0}}}\ln z_{-}+{\displaystyle \frac{2}{\unicode[STIX]{x03C0}}}\ln z_{+}\right]={\displaystyle \frac{H(t)}{2\unicode[STIX]{x03C0}}}\ln \unicode[STIX]{x1D6FD}tr={\displaystyle \frac{H(t)}{2\unicode[STIX]{x03C0}}}[\ln r+\ln \unicode[STIX]{x1D6FD}t]\end{eqnarray}$$
because, according to (1.5),
$z_{\pm }=\sqrt{\unicode[STIX]{x1D6FD}tr\text{e }^{\pm \text{i}\unicode[STIX]{x1D703}}}$
. The part that only depends on time
$t$
plays no role in getting
$\unicode[STIX]{x1D6FF}(t)\unicode[STIX]{x1D6FF}(x)\unicode[STIX]{x1D6FF}(y)$
from
$\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D6FB}^{2}G$
and therefore (1.3) has the correct behaviour near the origin. Finally, on the
$x$
axis in the limit
$\unicode[STIX]{x1D703}\rightarrow 0$
(east) we have
$z_{+}=z_{-}=\sqrt{\unicode[STIX]{x1D6FD}tr}$
and it is seen with (1.3) that for
$t>0$
(2.3a
) is recovered. For
$\unicode[STIX]{x1D703}\rightarrow \unicode[STIX]{x03C0}$
(west) we have
$z_{\pm }=\pm \text{i}\sqrt{\unicode[STIX]{x1D6FD}tr}$
and connection formulas are needed. They imply that
$\text{J}_{0}(z_{\pm })\rightarrow \text{I}_{0}(\sqrt{\unicode[STIX]{x1D6FD}tr})$
and that
$$\begin{eqnarray}\unicode[STIX]{x1D703}\rightarrow \unicode[STIX]{x03C0}:\quad \text{Y}_{0}(z_{+})\rightarrow \text{i}\text{I}_{0}(\sqrt{\unicode[STIX]{x1D6FD}tr})-{\displaystyle \frac{2}{\unicode[STIX]{x03C0}}}\text{K}_{0}(\sqrt{\unicode[STIX]{x1D6FD}tr}),\quad \text{Y}_{0}(z_{-})\rightarrow -\text{i}\text{I}_{0}(\cdots \,)-{\displaystyle \frac{2}{\unicode[STIX]{x03C0}}}\text{K}_{0}(\cdots \,)\end{eqnarray}$$
(see the NIST Handbook of Mathematical Functions (Olver et al.
Reference Olver, Lozier, Boisvert and Clark2010), formulas 10.27.6 and 10.27.11). Substituting this in (1.3) we find (2.3b
). The same result is found in the limit
$\unicode[STIX]{x1D703}=-\unicode[STIX]{x03C0}$
. This proves that
$G_{\unicode[STIX]{x1D6FF}}$
is correct.
3.2 Derivation and validation of
$G_{H}$
The integral representations showed that
$G_{H}$
can be obtained through differentiation of
$G_{\unicode[STIX]{x1D6FF}}$
according to (2.13a
). With the
$\{z_{\star },\bar{z}_{\star }\}$
variables we find that (2.13a
) becomes
$$\begin{eqnarray}G_{H}=-{\displaystyle \frac{4}{\unicode[STIX]{x1D6FD}(z_{\star }^{2}-{\bar{z}_{\star }}^{2})}}\left[z_{\star }{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z_{\star }}}-\bar{z}_{\star }{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\bar{z}_{\star }}}\right]G_{\unicode[STIX]{x1D6FF}}.\end{eqnarray}$$
Substitution of
$G_{\unicode[STIX]{x1D6FF}}=(1/4)\left[\text{J}_{0}(az_{\star })\text{Y}_{0}(a\bar{z}_{\star })+\text{c.c.}\right]$
results in the expression (1.4) for
$G_{H}$
with the definition of
$z_{\pm }$
given in (3.7) and use of the fact that
$\text{J}_{0}^{\prime }=-\text{J}_{1}$
and
$\text{Y}_{0}^{\prime }=-\text{Y}_{1}$
. Also the integral representations implied
$\unicode[STIX]{x2202}_{t}G_{H}=G_{\unicode[STIX]{x1D6FF}}$
. According to (3.17) this is true if
$$\begin{eqnarray}G_{\unicode[STIX]{x1D6FF}}=-{\displaystyle \frac{2t}{z_{+}^{2}-z_{-}^{2}}}\left[z_{+}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z_{+}}}-z_{-}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z_{-}}}\right]{\displaystyle \frac{\unicode[STIX]{x2202}G_{\unicode[STIX]{x1D6FF}}}{\unicode[STIX]{x2202}t}},\end{eqnarray}$$
having restored
$z_{\pm }$
on the right-hand side. We find, for example, that
$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}\text{J}_{0}(z_{+})\text{Y}_{0}(z_{-})}{\unicode[STIX]{x2202}t}}=\left[z_{+}\text{J}_{0}^{\prime }(z_{+})\text{Y}_{0}(z_{-})+z_{-}\text{J}_{0}(z_{+})\text{Y}_{0}^{\prime }(z_{-})\right]/2t\quad \text{and}\\ \displaystyle \left[z_{+}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z_{+}}}-z_{-}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}z_{-}}}\right]{\displaystyle \frac{\unicode[STIX]{x2202}\text{J}_{0}(z_{+})\text{Y}_{0}(z_{-})}{\unicode[STIX]{x2202}t}}=-{\displaystyle \frac{z_{+}^{2}-z_{-}^{2}}{2t}}\text{J}_{0}(z_{+})\text{Y}_{0}(z_{-})\end{array}\right\} & & \displaystyle\end{eqnarray}$$
after use of the Bessel equation (3.13). It quickly follows with (1.3) for
$G_{\unicode[STIX]{x1D6FF}}$
that indeed
$\unicode[STIX]{x2202}_{t}G_{H}=G_{\unicode[STIX]{x1D6FF}}$
. Moreover, since we know
${\mathcal{L}}G_{\unicode[STIX]{x1D6FF}}=0$
for
$r\neq 0$
and therefore
$\unicode[STIX]{x2202}_{t}{\mathcal{L}}G_{H}=0$
, it follows that
${\mathcal{L}}G_{H}=0$
(
${\mathcal{L}}G_{H}=f(x,y)\neq 0$
can be ruled out).
The singularity at the origin has to come from
$\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D6FB}^{2}G_{H}$
in (1.2b
) and the factor
$H(t)$
on the right-hand side must be due to a time derivative of
$tH(t)$
. Therefore for small
$r$
the Green’s function must behave as
$G_{H}\approx tH(t)\ln r/2\unicode[STIX]{x03C0}$
. That this is true follows with (3.14) for
$\text{J}_{0},\text{Y}_{0}$
and the fact that
$$\begin{eqnarray}z\rightarrow 0:\quad \text{J}_{1}(z)\sim z/2,\quad \text{Y}_{1}(z)\sim -2/(\unicode[STIX]{x03C0}z)+(z/\unicode[STIX]{x03C0})\ln z.\end{eqnarray}$$
When put in (1.4), we find that for small
$r$
$$\begin{eqnarray}r\downarrow 0:\quad G_{H}\approx {\displaystyle \frac{tH(t)}{2\unicode[STIX]{x03C0}}}[\ln r+\ln \unicode[STIX]{x1D6FD}t].\end{eqnarray}$$
Again the part that only depends on
$t$
is irrelevant, and
$G_{H}$
has the correct singularity.
Finally, it is easily verified with the Bessel equation (3.13) that
$\unicode[STIX]{x2202}_{t}$
(2.4a
)
$=$
(2.3a
). Also one finds
$\unicode[STIX]{x2202}_{t}$
(2.4b
)
$=$
(2.3b
) through use of
$\text{I}_{0}^{\prime }=\text{I}_{1}$
,
$\text{K}_{0}^{\prime }=-\text{K}_{1}$
and the fact that
$\text{I}_{0}(z)$
satisfies
$z^{2}{\mathcal{I}}_{0}^{\prime \prime }+z{\mathcal{I}}_{0}^{\prime }-z^{2}{\mathcal{I}}_{0}=0$
and
$\text{K}_{0}(z)$
too. Since we have just shown that everywhere
$\unicode[STIX]{x2202}_{t}G_{H}=G_{\unicode[STIX]{x1D6FF}}$
and
$G_{\unicode[STIX]{x1D6FF}}$
given by (1.3) has the correct limiting behaviour on the east–west axis, our expression for
$G_{H}$
(1.4) reduces there to (2.4a
) and (2.4b
). This has been verified independently with a Taylor series expansion of
$G_{H}$
about
$\unicode[STIX]{x1D703}=0$
and
$\unicode[STIX]{x1D703}=\pm \unicode[STIX]{x03C0}$
.
4 Properties of the Green’s functions
In the integral expressions (2.1) and (2.5a
) for
$G_{\unicode[STIX]{x1D6FF}}$
and (2.2) and (2.5b
) for
$G_{H}$
, the arguments of the integrands contain the similarity variable
$\unicode[STIX]{x1D6FD}tr$
. Our new expressions for
$G_{\unicode[STIX]{x1D6FF}}$
(1.3) and for
$G_{H}$
(1.4) also contain this variable via the complex conjugate variables
$z_{\pm }$
defined in (1.5). If time
$t$
and distance
$r$
were dimensional, the combination
$\unicode[STIX]{x1D6FD}tr$
is a non-dimensional variable. The mathematical problem has however already been formulated with non-dimensional time
$t$
and
$r$
or
$\{x,y\}$
through scaling with some time scale
$T$
and length scale
$L$
, i.e. in (1.2a–c
) the variables are actually non-dimensional
$\tilde{t}=t/T$
and
$\{\tilde{x},{\tilde{y}}\}=\{x/L,y/L\}$
and non-dimensionally
$\tilde{\unicode[STIX]{x1D6FD}}=\unicode[STIX]{x1D6FD}\times TL$
. For convenience we have however dropped the tildes. In the graphs below we shall plot the response against non-dimensional
$\{x,y\}$
and time
$t$
scaled with arbitrary
$L$
and
$T$
but in the numerical evaluations we have set
$\unicode[STIX]{x1D6FD}=1$
.
4.1 Response to impulsive forcing:
$G_{\unicode[STIX]{x1D6FF}}$
Graphing the exact solution
$G_{\unicode[STIX]{x1D6FF}}$
given in (1.3) is easy with available numerical packages although some care is required on the western axis where
$\unicode[STIX]{x1D703}=\pm \unicode[STIX]{x03C0}$
. Due to ambiguity as to how the Bessel functions behave as their arguments become purely imaginary (
$z_{\pm }=\pm \text{i}\sqrt{\unicode[STIX]{x1D6FD}tr}$
), it is best to use a numerical grid that has no points exactly on the western axis. Since spatially the response depends only on the similarity coordinates
$\unicode[STIX]{x1D6FD}tx$
,
$\unicode[STIX]{x1D6FD}ty$
or
$\unicode[STIX]{x1D6FD}tr$
and
$\unicode[STIX]{x1D703}$
, for any
$t=t_{0}>0$
all information is contained in the response at
$t_{0}$
. But it is illuminating to graph the response as a function of time. As mentioned in the introduction, very little effort is required to create a movie that vividly illustrates the evolution.
Contours of
$G_{\unicode[STIX]{x1D6FF}}$
(1.3) at (a) non-dimensional time
$t=0.25$
and (b)
$t=0.5$
. White circle in (a) has a radius
$r=50$
and in (b) a radius
$r=25$
. Because of self-similarity in (b) at the double the time compared to (a), the pattern within the circle is exactly the same. Black dot ● at the centre
$r=0$
indicates the source origin. White space thereabouts indicates extreme negative values associated with the logarithmic singularity of
$G_{\unicode[STIX]{x1D6FF}}$
(see text). In (a) the black dashed line is the parabola
$2a\unicode[STIX]{x1D701}=\unicode[STIX]{x03C0}/2$
with
$\unicode[STIX]{x1D701}=\sqrt{x+r}$
and
$a=\sqrt{\unicode[STIX]{x1D6FD}t/2}$
. Along this contour
$G_{\unicode[STIX]{x1D6FF}}=0$
. In (b) the white dashed parabolas are
$2a\unicode[STIX]{x1D701}=(n+1)\unicode[STIX]{x03C0}$
with
$n=0,1,2$
.
Thus in figure 3 we chose non-dimensional time
$t=t_{0}=0.25$
(a) and
$t=2t_{0}=0.5$
(b). In (a) we have drawn a circle with radius
$r_{0}=50$
and in (b) with radius
$r=(1/2)r_{0}=25$
. Therefore in both panels the circles have
$tr=t_{0}r_{0}$
and the self-similarity is evident: in figure 3(b) the same pattern is seen within the circle as in figure 3(a) but shrunk to half the scale. For increasing time more of the ‘banana’ shaped regions appear to propagate from the east towards the source location indicated by ‘●’. An ever larger number gets wrapped about the origin, with increasing curvature.
West of the source a wake-like region exists forever concentrated about the western axis. In figure 3(a) the boundary of the wake is indicated by the dashed curve along which
$G_{\unicode[STIX]{x1D6FF}}=0$
. Within this region, all streamlines indicate flow with the same sense of circulation about the origin but elongated in the western direction. East of the source, contours near the east–west axis are nearly north–south thus indicating predominantly north–south motions with velocity
$v$
advecting planetary vorticity. In the white area in figure 3 near the source location, the response exceeds some arbitrarily chosen (negative) amplitude threshold. Veronis (Reference Veronis1958) was already able to sketch these patterns by numerically solving a partial differential equation similar to (2.13b
) (see also Longuet-Higgins Reference Longuet-Higgins1965). Near the source location (the white region in figure 3), according to (3.15) the singular behaviour persists.
With the known properties
$$\begin{eqnarray}z\rightarrow \infty :\quad \text{J}_{\unicode[STIX]{x1D708}}(z)\sim \sqrt{2/(\unicode[STIX]{x03C0}z)}\cos (z-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D708}\unicode[STIX]{x03C0}-{\textstyle \frac{1}{4}}\unicode[STIX]{x03C0}),\quad \text{Y}_{\unicode[STIX]{x1D708}}(z)\sim \sqrt{2/(\unicode[STIX]{x03C0}z)}\sin (z-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D708}\unicode[STIX]{x03C0}-{\textstyle \frac{1}{4}}\unicode[STIX]{x03C0}),\end{eqnarray}$$
according to (1.3)
$$\begin{eqnarray}\unicode[STIX]{x1D6FD}tr\rightarrow \infty :\quad G_{\unicode[STIX]{x1D6FF}}\sim -{\displaystyle \frac{\cos (z_{+}+z_{-})}{2\unicode[STIX]{x03C0}\sqrt{z_{+}z_{-}}}}.\end{eqnarray}$$
With the definitions of
$z_{\pm }$
in (3.7) and the parabolic coordinates
$\{\unicode[STIX]{x1D701},\unicode[STIX]{x1D702}\}$
in (3.8):
$$\begin{eqnarray}\unicode[STIX]{x1D6FD}tr\rightarrow \infty :\quad G_{\unicode[STIX]{x1D6FF}}\sim -{\displaystyle \frac{\cos (2a\unicode[STIX]{x1D701})}{2\unicode[STIX]{x03C0}\sqrt{a^{2}(\unicode[STIX]{x1D701}^{2}+\unicode[STIX]{x1D702}^{2})}}}=-{\displaystyle \frac{\cos (\sqrt{2\unicode[STIX]{x1D6FD}t(x+r)})}{2\unicode[STIX]{x03C0}\sqrt{\unicode[STIX]{x1D6FD}tr}}}\equiv G_{\unicode[STIX]{x1D6FF}}^{a}.\end{eqnarray}$$
Lines of constant
$\unicode[STIX]{x1D701}=\sqrt{x+r}$
are parabolas that open towards the west (
$x<0$
) as shown in figure 2. Some of these parabolas have been drawn in figure 3 as dashed curves. Longuet-Higgins (Reference Longuet-Higgins1965) already predicted that lines of constant phase would be such parabolas. The approximation
$G_{\unicode[STIX]{x1D6FF}}^{a}$
in (4.3) was recently found by Webb, Duba & Hu (Reference Webb, Duba and Hu2016), but as the large-time behaviour implied by an (inverse Laplace transform) integral representation of
$G_{\unicode[STIX]{x1D6FF}}$
.
The approximation (4.3) predicts zeros for
$2a\unicode[STIX]{x1D701}=(n+1/2)\unicode[STIX]{x03C0}$
(
$n=0,1,\ldots$
). The first zero (
$n=0$
) has
$2a\unicode[STIX]{x1D701}=\unicode[STIX]{x03C0}/2$
which has been drawn in figure 3(a) as the dashed black parabola, delineating the ‘wake’. The approximation predicts on the eastern side maxima/minima for
$2a\unicode[STIX]{x1D701}=(n+1)\unicode[STIX]{x03C0}$
. For
$n=0,1,2$
these are the three white dashed parabolas drawn in figure 3(b). It is seen that they form the ‘spine’ of the curved patterns.
The approximation reveals the salient behaviour for large
$\unicode[STIX]{x1D6FD}tr$
that is hidden in the exact but complex expression (1.3) for
$G_{\unicode[STIX]{x1D6FF}}$
. For large
$\unicode[STIX]{x1D6FD}tr$
or
$\unicode[STIX]{x1D6FD}t|x|$
on the east–west axis
$$\begin{eqnarray}\unicode[STIX]{x1D703}=0\,(\text{east}):\quad ~G_{\unicode[STIX]{x1D6FF}}^{a}=-{\displaystyle \frac{\cos (2\sqrt{\unicode[STIX]{x1D6FD}tr})}{2\unicode[STIX]{x03C0}\sqrt{\unicode[STIX]{x1D6FD}tr}}},\quad \unicode[STIX]{x1D703}=\pm \unicode[STIX]{x03C0}\,(\text{west}):\quad G_{\unicode[STIX]{x1D6FF}}^{a}=-{\displaystyle \frac{1}{2\unicode[STIX]{x03C0}\sqrt{\unicode[STIX]{x1D6FD}tr}}}.\end{eqnarray}$$
In figure 4 we compare the approximation on this axis with the known exact solution(s) (2.3a,b
). This cross-section along the axis illustrates some of the features seen in figure 3. In particular, the first zero crossing with
$2a\unicode[STIX]{x1D701}=2\sqrt{\unicode[STIX]{x1D6FD}tx}=(1/2)\unicode[STIX]{x03C0}$
on the eastern side corresponds to the ‘wake’ defined by
$G_{\unicode[STIX]{x1D6FF}}=0$
(the black parabola in figure 3
a).
Graph showing the exact
$G_{\unicode[STIX]{x1D6FF}}$
(1.3) on the east–west axis through the source and the approximation
$G_{\unicode[STIX]{x1D6FF}}^{a}$
(4.4a,b
). According to (2.3a
) to the east:
$G_{\unicode[STIX]{x1D6FF}}=\text{J}_{0}(\unicode[STIX]{x1D70C})\text{Y}_{0}(\unicode[STIX]{x1D70C})/2$
and to the west according to (2.3b
)
$G_{\unicode[STIX]{x1D6FF}}=-\text{I}_{0}(\unicode[STIX]{x1D70C})\text{K}_{0}(\unicode[STIX]{x1D70C})/\unicode[STIX]{x03C0}$
with
$\unicode[STIX]{x1D70C}=\sqrt{\unicode[STIX]{x1D6FD}t|x|}$
. For convenience we used
$\unicode[STIX]{x1D6FD}=1$
.
4.2 Response to sustained forcing:
$G_{H}$
In figure 5 we show contours of the exact solution
$G_{H}$
given in (1.4) for non-dimensional times
$t=t_{0}=0.5$
(a) and
$t=2t_{0}=1$
. As in figure 3 for
$G_{\unicode[STIX]{x1D6FF}}$
, we have drawn in (a) a circle with radius
$r_{0}=50$
and in (b) with radius
$r=(1/2)r_{0}=25$
. In both panels the circles have
$tr=t_{0}r_{0}$
and spatial self-similarity is evident. The source location is again indicated by ‘●’. Close to the source according to (3.21) with increasing time the singularity grows in amplitude, if this can be said in regard to something that has infinite amplitude. Within the white areas in figure 5,
$G_{H}$
exceeds some (negative) amplitude threshold associated with this singularity.
Contours of
$G_{H}$
at (a) non-dimensional time
$t=0.5$
and (b)
$t=1.0$
. White circle in (a) has a radius
$r=50$
and in (b) a radius
$r=25$
. In (a) the black dashed line is the parabola
$2a\unicode[STIX]{x1D701}=\unicode[STIX]{x03C0}$
with
$\unicode[STIX]{x1D701}=\sqrt{x+r}$
and
$a=\sqrt{\unicode[STIX]{x1D6FD}t/2}$
. Along this curve
$G_{H}=0$
. In (b) the white dashed lines are for
$2a\unicode[STIX]{x1D701}=(n+3/2)\unicode[STIX]{x03C0}$
with
$n=0,1$
.
According to (4.1), for large arguments
$|z_{\pm }|\gg 1$
, the terms within square brackets in (1.4) become
$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle z_{+}[\text{J}_{1}(z_{+})\text{Y}_{0}(z_{-})+\text{J}_{0}(z_{-})\text{Y}_{1}(z_{+})]\sim -{\displaystyle \frac{2}{\unicode[STIX]{x03C0}}}{\displaystyle \frac{z_{+}}{\sqrt{z_{+}z_{-}}}}\sin (z_{+}+z_{-}),\\ \displaystyle z_{-}[\text{J}_{1}(z_{-})\text{Y}_{0}(z_{+})+\text{J}_{0}(z_{+})\text{Y}_{1}(z_{-})]\sim -{\displaystyle \frac{2}{\unicode[STIX]{x03C0}}}{\displaystyle \frac{z_{-}}{\sqrt{z_{+}z_{-}}}}\sin (z_{+}+z_{-}).\end{array}\right\}\end{eqnarray}$$
$$\begin{eqnarray}\unicode[STIX]{x1D6FD}tr\rightarrow \infty :\quad G_{H}\sim -{\displaystyle \frac{1}{\unicode[STIX]{x03C0}}}{\displaystyle \frac{t\sin (z_{+}+z_{-})}{(z_{+}+z_{-})\sqrt{z_{+}z_{-}}}}\end{eqnarray}$$
and with (3.7), equation (3.8)
$$\begin{eqnarray}\unicode[STIX]{x1D6FD}tr\rightarrow \infty :\quad G_{H}\sim -{\displaystyle \frac{1}{\unicode[STIX]{x03C0}}}{\displaystyle \frac{\sin (2a\unicode[STIX]{x1D701})}{\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D701}\sqrt{\unicode[STIX]{x1D701}^{2}+\unicode[STIX]{x1D702}^{2}}}}=-{\displaystyle \frac{\sin (\sqrt{2\unicode[STIX]{x1D6FD}t(x+r)})}{\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FD}\sqrt{x+r}\sqrt{2r}}}\equiv G_{H}^{a}.\end{eqnarray}$$
This agrees with the result of Kamenkovich (Reference Kamenkovich1989) who derived large time expansions of
$G_{H}$
via a consideration of the asymptotic properties of inverse Laplace transforms. There is a noticeable difference between
$G_{\unicode[STIX]{x1D6FF}}$
and
$G_{H}$
: whereas according to (4.3) amplitudes of
$G_{\unicode[STIX]{x1D6FF}}$
decay with time as
$t^{-1/2}$
, amplitudes of
$G_{H}$
are constant (with the exception of the west axis; see below). Nonetheless
$G_{\unicode[STIX]{x1D6FF}}=\unicode[STIX]{x2202}G_{H}/\unicode[STIX]{x2202}t$
and one easily verifies with (4.3) and (4.7) that also
$G_{\unicode[STIX]{x1D6FF}}^{a}=\unicode[STIX]{x2202}G_{H}^{a}/\unicode[STIX]{x2202}t$
.
The approximation (4.7) predicts zeros for
$2a\unicode[STIX]{x1D701}=(n+1)\unicode[STIX]{x03C0}$
(
$n=0,1,\ldots$
) and again
$\unicode[STIX]{x1D701}=\sqrt{x+r}$
and
$a=\sqrt{\unicode[STIX]{x1D6FD}t/2}$
. The first zero has
$2a\unicode[STIX]{x1D701}=\unicode[STIX]{x03C0}$
and the ‘wake’ defined by this curve on which
$G_{H}=0$
is the black dashed line in figure 5(a). The approximation predicts on the eastern side maxima/minima for
$2a\unicode[STIX]{x1D701}=(n+3/2)\unicode[STIX]{x03C0}$
. The first two (
$n=0,1$
) are the dashed parabolas drawn in figure 5(b).
Graph showing the exact
$G_{H}$
(1.4) on the east–west axis through the source and the approximation
$G_{H}^{a}$
(4.8a,b
) at non-dimensional time
$t=0.5$
. According to (2.4a
) to the east:
$G_{H}=t[\text{J}_{0}(\unicode[STIX]{x1D70C})\text{Y}_{0}(\unicode[STIX]{x1D70C})+\text{J}_{1}(\unicode[STIX]{x1D70C})\text{Y}_{1}(\unicode[STIX]{x1D70C})]/2$
and to the west according to (2.4b
)
$G_{H}=-t[\text{I}_{0}(\unicode[STIX]{x1D70C})\text{K}_{0}(\unicode[STIX]{x1D70C})+\text{I}_{1}(\unicode[STIX]{x1D70C})\text{K}_{1}(\unicode[STIX]{x1D70C})]/\unicode[STIX]{x03C0}$
with argument
$\unicode[STIX]{x1D70C}=\sqrt{\unicode[STIX]{x1D6FD}t|x|}$
.
Comparison of figure 5 with figure 3 reveals a difference: in figure 5 the response to the east of the forcing appears to contain fewer of the typical patterns. Why this is can be illustrated with the far field approximation of
$G_{H}$
given by (4.7), i.e. for large
$\unicode[STIX]{x1D6FD}tr$
on the east–west axis:
$$\begin{eqnarray}\unicode[STIX]{x1D703}=0\,(\text{east}):\quad G_{H}^{a}=-{\displaystyle \frac{\sin (2\sqrt{\unicode[STIX]{x1D6FD}tr})}{2\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FD}r}},\quad \unicode[STIX]{x1D703}=\pm \unicode[STIX]{x03C0}\,(\text{west}):\quad G_{H}^{a}=-{\displaystyle \frac{1}{\unicode[STIX]{x03C0}}}\sqrt{{\displaystyle \frac{t}{\unicode[STIX]{x1D6FD}r}}}.\end{eqnarray}$$
In figure 6 we compare the approximation on the entire axis with the exact solution(s) (2.4a,b
). This explains why in figure 5 there are fewer of the parabolically shaped regions visible east of the forcing: on the eastern axis for fixed time (4.4a
) reveals that
$G_{\unicode[STIX]{x1D6FF}}\propto 1/\sqrt{r}$
whereas (4.8a
) indicates
$G_{H}\propto 1/r$
, i.e.
$G_{H}$
decays more rapidly towards the east than
$G_{\unicode[STIX]{x1D6FF}}$
with distance
$r$
from the source location. The two prominent patterns east of the source seen in figure 5(a) correspond to the first maximum and minimum in the cross-section shown in figure 6, i.e.
$2a\unicode[STIX]{x1D701}=2\sqrt{\unicode[STIX]{x1D6FD}tx}=(n+3/2)\unicode[STIX]{x03C0}$
and
$n=0,1$
. Finally, note that (4.6) shows that amplitudes do not decay or grow with time
$t$
except exactly on the west axis (
$\unicode[STIX]{x1D701}=0$
) where
$$\begin{eqnarray}\unicode[STIX]{x1D701}\downarrow 0:\quad {\displaystyle \frac{\sin (2a\unicode[STIX]{x1D701})}{\unicode[STIX]{x1D701}}}\rightarrow 2a=\sqrt{2\unicode[STIX]{x1D6FD}t}\end{eqnarray}$$
and this singular limit leads to the unbounded growth of
$G_{H}$
displayed in (4.8b
).
4.3 Kinetic energy distributions
The approximation (4.3) suggests that at a fixed location with increasing time, the quasi-wave field associated with
$G_{\unicode[STIX]{x1D6FF}}$
disappears with
$1/\sqrt{t}$
. Thus, nothing but the singularity (3.15) would remain. But a more relevant quantity is kinetic energy
$E$
. This is determined by gradients of the streamfunction
$\unicode[STIX]{x1D713}=G_{\unicode[STIX]{x1D6FF}}$
and because with increasing time the spatial scales decrease, gradients increase. This compensates for the overall decrease of the amplitude of
$G_{\unicode[STIX]{x1D6FF}}$
.
Contours of kinetic energy
$E_{\unicode[STIX]{x1D6FF}}$
given by (4.11) at
$t=0.5$
. The innermost dashed parabola is defined by
$2a\unicode[STIX]{x1D701}=(n+1/2)\unicode[STIX]{x03C0}$
with
$n=0$
and
$\unicode[STIX]{x1D701}=\sqrt{x+r}$
,
$a=\sqrt{\unicode[STIX]{x1D6FD}t/2}$
. This corresponds to the ‘wake’ boundary shown in figure 3(a) along which
$G_{\unicode[STIX]{x1D6FF}}=0$
. The second parabola is for
$n=1$
. Energy along the circle with radius
$r=40$
is shown in figure 8.
The kinetic energy associated with the Green’s function is
$E=(1/2)\unicode[STIX]{x1D735}G\boldsymbol{\cdot }\unicode[STIX]{x1D735}G$
. The easiest way to calculate
$E$
is to use the fact that in parabolic coordinates
$$\begin{eqnarray}\unicode[STIX]{x1D735}={\displaystyle \frac{1}{\sqrt{\unicode[STIX]{x1D701}^{2}+\unicode[STIX]{x1D702}^{2}}}}\left[\hat{\boldsymbol{\unicode[STIX]{x1D701}}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}}+\hat{\boldsymbol{\unicode[STIX]{x1D702}}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}}\right],\end{eqnarray}$$
with
$\{\hat{\boldsymbol{\unicode[STIX]{x1D701}}},\hat{\boldsymbol{\unicode[STIX]{x1D702}}}\}$
the orthogonal unit vectors associated with
$\{\unicode[STIX]{x1D701},\unicode[STIX]{x1D702}\}$
(see Morse & Feshbach Reference Morse and Feshbach1953; Moon & Spencer Reference Moon and Spencer1961). Without going into any further details, we find
$$\begin{eqnarray}\displaystyle E & = & \displaystyle {\displaystyle \frac{1}{2(\unicode[STIX]{x1D701}^{2}+\unicode[STIX]{x1D702}^{2})}}\left[\left({\displaystyle \frac{\unicode[STIX]{x2202}G}{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}}\right)^{2}+\left({\displaystyle \frac{\unicode[STIX]{x2202}G}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}}\right)^{2}\right]\nonumber\\ \displaystyle & = & \displaystyle \left({\displaystyle \frac{1}{4}}\right)^{2}{\displaystyle \frac{\unicode[STIX]{x1D6FD}t}{\unicode[STIX]{x1D701}^{2}+\unicode[STIX]{x1D702}^{2}}}|\text{J}_{0}(z_{+})\text{Y}_{1}(z_{-})+\text{J}_{1}(z_{-})\text{Y}_{0}(z_{+})|^{2}=E_{\unicode[STIX]{x1D6FF}}\end{eqnarray}$$
when
$G=G_{\unicode[STIX]{x1D6FF}}$
given by (1.3) is substituted in (4.11). This may give the impression that the kinetic energy grows with time
$t$
. But, for large
$\unicode[STIX]{x1D6FD}tr$
it follows with (4.1) that
$$\begin{eqnarray}\unicode[STIX]{x1D6FD}tr\rightarrow \infty :\quad \text{J}_{0}(z_{+})\text{Y}_{1}(z_{-})+\text{J}_{1}(z_{-})\text{Y}_{0}(z_{+})\sim -{\displaystyle \frac{2}{\unicode[STIX]{x03C0}}}{\displaystyle \frac{\sin (z_{+}+z_{-})}{\sqrt{z_{+}z_{-}}}}.\end{eqnarray}$$
Squaring (4.12) and substitution in (4.11) yields for large
$\unicode[STIX]{x1D6FD}tr$
the approximation
$$\begin{eqnarray}E_{\unicode[STIX]{x1D6FF}}\sim {\displaystyle \frac{1}{2}}{\displaystyle \frac{\sin ^{2}(2a\unicode[STIX]{x1D701})}{\unicode[STIX]{x03C0}^{2}(\unicode[STIX]{x1D701}^{2}+\unicode[STIX]{x1D702}^{2})^{2}}}={\displaystyle \frac{1}{2}}{\displaystyle \frac{\sin ^{2}(\sqrt{2\unicode[STIX]{x1D6FD}t(x+r)})}{(2\unicode[STIX]{x03C0}r)^{2}}}\equiv E_{\unicode[STIX]{x1D6FF}}^{a}.\end{eqnarray}$$
In figure 7 we show contours of kinetic energy
$E_{\unicode[STIX]{x1D6FF}}$
(4.11). The patterns are similar to that of
$G_{\unicode[STIX]{x1D6FF}}$
shown in figure 3. The one-term approximation (4.13) predicts maximal energy along parabolas
$2a\unicode[STIX]{x1D701}=(n+1/2)\unicode[STIX]{x03C0}$
with
$n=0,1,\ldots$
(
$\unicode[STIX]{x1D701}=\sqrt{x+r}$
,
$a=\sqrt{\unicode[STIX]{x1D6FD}t/2}$
). In figure 7 the innermost dashed parabola is for
$n=0$
and coincides with the wake boundary shown in figure 3(a). This is a location of maximal kinetic energy. In figure 7 the second parabola (
$n=1$
) is seen to match the next pattern of maximal energy well. In figure 7 we have drawn a circle with radius
$r=40$
. Going around this circle of radius
$r=40$
the peaks of energy are shown in figure 8 as well as for a circle of smaller radius
$r=30$
. The approximation (4.13), based on the assumption
$\unicode[STIX]{x1D6FD}tr\gg 1$
, shows that at large
$r$
all maxima will have the same amplitude, as observed in figure 8. In polar coordinates the approximation
$$\begin{eqnarray}E_{\unicode[STIX]{x1D6FF}}^{a}={\displaystyle \frac{1}{2}}{\displaystyle \frac{\sin ^{2}(2a\unicode[STIX]{x1D701})}{(2\unicode[STIX]{x03C0}r)^{2}}},\quad 2a\unicode[STIX]{x1D701}=\sqrt{2\unicode[STIX]{x1D6FD}tr}\sqrt{1+\cos \unicode[STIX]{x1D703}}\end{eqnarray}$$
shows that the energy peaks decay with distance
$1/r^{2}$
from the source and the peak positions in terms of the polar angle
$\unicode[STIX]{x1D703}$
are determined by
$2\unicode[STIX]{x1D6FD}tr(1+\cos \unicode[STIX]{x1D703})=(n+1/2)^{2}\unicode[STIX]{x03C0}^{2}$
. At fixed radius
$r$
with increasing time
$t$
more peaks appear as continuously more of the parabolic patterns seen in figure 3 coming from the eastern side get wrapped about the forcing origin.
The energy associated with
$G_{H}$
can be calculated exactly too and we find
$$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}G_{H}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}}={\displaystyle \frac{1}{\unicode[STIX]{x1D702}}}\left\{t(G_{\unicode[STIX]{x1D6FF}}+{\textstyle \frac{1}{4}}[\text{J}_{1}(z_{+})\text{Y}_{1}(z_{-})+\text{c.c.}])-G_{H}\right\}, & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}G_{H}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}}={\displaystyle \frac{1}{\unicode[STIX]{x1D701}}}\{t(G_{\unicode[STIX]{x1D6FF}}-{\textstyle \frac{1}{4}}[\text{J}_{1}(z_{+})\text{Y}_{1}(z_{-})+\text{c.c.}])-G_{H}\}, & \displaystyle\end{eqnarray}$$
$G_{\unicode[STIX]{x1D6FF}}$
as in (1.3) and
$G_{H}$
again given by (1.4). Division by
$\unicode[STIX]{x1D702}$
in (4.15a
) may be cause of concern at first sight, but
$\unicode[STIX]{x1D702}=0$
corresponds to the east axis and there according to (2.3a
) and (2.4a
): 
$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D702}=0~\text{(east)}:\quad G_{\unicode[STIX]{x1D6FF}}={\textstyle \frac{1}{2}}\text{J}_{0}(\unicode[STIX]{x1D70C})\text{Y}_{0}(\unicode[STIX]{x1D70C}),\quad G_{H}={\textstyle \frac{1}{2}}t[\text{J}_{0}(\unicode[STIX]{x1D70C})\text{Y}_{0}(\unicode[STIX]{x1D70C})+\text{J}_{1}(\unicode[STIX]{x1D70C})\text{Y}_{1}(\unicode[STIX]{x1D70C})],\\ \displaystyle \text{and}\quad {\textstyle \frac{1}{4}}[\text{J}_{1}(z_{+})\text{Y}_{1}(z_{-})+\text{c.c.}]={\textstyle \frac{1}{2}}\text{J}_{1}(\unicode[STIX]{x1D70C})\text{Y}_{1}(\unicode[STIX]{x1D70C}),\quad \unicode[STIX]{x1D70C}=\sqrt{\unicode[STIX]{x1D6FD}tr}\end{array}\right\}\end{eqnarray}$$
so that in the limit
$\unicode[STIX]{x1D702}=0$
the numerator of (4.15a
) vanishes and an expansion in small
$\unicode[STIX]{x1D702}$
can be shown to give a finite answer for
$\unicode[STIX]{x2202}G_{H}/\unicode[STIX]{x2202}\unicode[STIX]{x1D702}$
. Also the limit
$\unicode[STIX]{x1D701}=0$
appears dangerous in (4.15b
) but again a finite answer is obtained by taking into account that there
$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\unicode[STIX]{x1D701}=0~\text{(west)}:\quad G_{\unicode[STIX]{x1D6FF}}=-{\displaystyle \frac{\text{I}_{0}(\unicode[STIX]{x1D70C})\text{K}_{0}(\unicode[STIX]{x1D70C})}{\unicode[STIX]{x03C0}}},\quad G_{H}=-{\displaystyle \frac{t}{\unicode[STIX]{x03C0}}}\left[\text{I}_{0}(\unicode[STIX]{x1D70C})\text{K}_{0}(\unicode[STIX]{x1D70C})+\text{I}_{1}(\unicode[STIX]{x1D70C})\text{K}_{1}(\unicode[STIX]{x1D70C})\right],\\ \frac{1}{4}[\text{J}_{1}(z_{+})\text{Y}_{1}(z_{-})+\text{c.c.}]={\displaystyle \frac{\text{I}_{1}(\unicode[STIX]{x1D70C})\text{K}_{1}(\unicode[STIX]{x1D70C})}{\unicode[STIX]{x03C0}}},\quad \unicode[STIX]{x1D70C}=\sqrt{\unicode[STIX]{x1D6FD}tr}.\end{array}\right\}\end{eqnarray}$$
Therefore in the limit
$\unicode[STIX]{x1D701}=0$
the numerator of (4.15b
) vanishes and
$\unicode[STIX]{x2202}G_{H}/\unicode[STIX]{x2202}\unicode[STIX]{x1D701}$
is well defined. Contours of
$$\begin{eqnarray}E_{H}={\displaystyle \frac{1}{2(\unicode[STIX]{x1D701}^{2}+\unicode[STIX]{x1D702}^{2})}}\left[\left({\displaystyle \frac{\unicode[STIX]{x2202}G_{H}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}}\right)^{2}+\left({\displaystyle \frac{\unicode[STIX]{x2202}G_{H}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}}\right)^{2}\right]\end{eqnarray}$$
are shown in figure 9. The energy along the circle with radius
$r=30$
drawn in figure 9 is shown in figure 10.
Contours of energy
$E_{H}$
according to (4.18) at
$t=1$
. The innermost dashed parabola is for
$2a\unicode[STIX]{x1D701}=\sqrt{2}$
with
$\unicode[STIX]{x1D701}=\sqrt{x+r}$
,
$a=\sqrt{\unicode[STIX]{x1D6FD}t/2}$
. This lies inside to the ‘wake’ boundary shown in figure 5(a) along which
$G_{H}=0$
and for which
$2a\unicode[STIX]{x1D701}=\unicode[STIX]{x03C0}$
. Energy along the circle with radius
$r=30$
is shown in figure 10.
Whereas it is easy to find the approximation (4.13) for the kinetic energy
$E_{\unicode[STIX]{x1D6FF}}$
by employing the asymptotic properties of
$\text{J}_{\unicode[STIX]{x1D708}},\text{Y}_{\unicode[STIX]{x1D708}}$
according to (4.1), things become quite difficult if the same route is taken here for
$E_{H}$
: it requires the rarely used higher-order corrections to (4.1). Instead we calculate the approximation by differentiation of (4.7). This yields in polar coordinates
$$\begin{eqnarray}\displaystyle E_{H} & {\sim} & \displaystyle {\displaystyle \frac{1}{2}}{\displaystyle \frac{1}{(2\unicode[STIX]{x03C0}r)^{2}}}\left({\displaystyle \frac{t}{\unicode[STIX]{x1D6FD}r}}\right){\displaystyle \frac{1}{1+\cos \unicode[STIX]{x1D703}}}\left[2\cos ^{2}(2a\unicode[STIX]{x1D701})-{\displaystyle \frac{2(3+\cos \unicode[STIX]{x1D703})\cos (2a\unicode[STIX]{x1D701})\sin (2a\unicode[STIX]{x1D701})}{(2a\unicode[STIX]{x1D701})}}\right.\nonumber\\ \displaystyle & & \displaystyle \left.+{\displaystyle \frac{(5+3\cos \unicode[STIX]{x1D703})\sin ^{2}(2a\unicode[STIX]{x1D701})}{(2a\unicode[STIX]{x1D701})^{2}}}\right]\equiv E_{H}^{a},\quad 2a\unicode[STIX]{x1D701}=\sqrt{2\unicode[STIX]{x1D6FD}t}\sqrt{r(1+\cos \unicode[STIX]{x1D703})}.\end{eqnarray}$$
For large time the first term within square brackets,
$\cos ^{2}(2a\unicode[STIX]{x1D701})$
, dominates but the next two terms cannot be discarded: they are both required to negate the singular behaviour of the prefactor
$1/r(1+\cos \unicode[STIX]{x1D703})=1/\unicode[STIX]{x1D701}^{2}$
. In other words, in the limit
$\unicode[STIX]{x1D703}=\pm \unicode[STIX]{x03C0}$
or
$\unicode[STIX]{x1D701}=0$
(the western axis) all three terms are required. In that limit
$\sin (2a\unicode[STIX]{x1D701})/(2a\unicode[STIX]{x1D701})\rightarrow 1$
and the bracketed term in (4.19) evaluates to
$$\begin{eqnarray}[2-2(3-1)+(5-3)]=0\end{eqnarray}$$
and expansions of the second and third term in powers of small
$\unicode[STIX]{x1D701}$
always yield a finite result. In figure 10 we compare the approximation (4.19) with the exact energy along the circle of radius
$r=30$
also shown in figure 9 and at a smaller radius
$r=25$
. We simply chose
$t=1$
which was also used for figure 5(b). Comparison of figures 9 and 10 with the corresponding figures 7 and 8 for
$G_{\unicode[STIX]{x1D6FF}}$
reveals that for
$G_{H}$
the kinetic energy is dominated by two plumes on the western side of the forcing. Another differences is that (4.19) shows that away from the western axis,
$E_{H}$
decays with distance from the forcing with
$1/r^{3}$
while linearly growing in time while (4.13) shows that
$E_{\unicode[STIX]{x1D6FF}}$
decays with
$1/r^{2}$
with amplitudes that are independent of time.
Finally, as in figure 7 for
$E_{\unicode[STIX]{x1D6FF}}$
we have drawn in figure 9 two parabolas
$\unicode[STIX]{x1D701}=\text{const.}$
that fit the maximal energy patterns. For
$G_{\unicode[STIX]{x1D6FF}}$
in figure 7 the innermost parabola coincided with the wake boundary
$G_{\unicode[STIX]{x1D6FF}}=0$
and
$2a\unicode[STIX]{x1D701}=(1/2)\unicode[STIX]{x03C0}$
: this is the first maximum of the leading-order term
$\sin ^{2}(2a\unicode[STIX]{x1D701})$
in (4.14). But for
$E_{H}$
the first maximum is not determined by
$\cos ^{2}(2a\unicode[STIX]{x1D701})$
with maxima at
$2a\unicode[STIX]{x1D701}=(n+1)\unicode[STIX]{x03C0}$
but by the combination of all three terms within the square brackets in (4.19). We find that the parabolic axis through the dominant energy pattern in figure 9 lies well within the wake region: in figure 5(a) the wake boundary (dashed parabola) is defined by
$2a\unicode[STIX]{x1D701}=\unicode[STIX]{x03C0}$
but in figure 9 the axis was found to coincide with
$2a\unicode[STIX]{x1D701}=\sqrt{2}<\unicode[STIX]{x03C0}$
and the second parabola with
$2a\unicode[STIX]{x1D701}=4<2\unicode[STIX]{x03C0}$
. At later times (not shown) the tendency is that the axis of the dominant energy ‘plumes’ of
$E_{H}$
moves further towards the west axis, i.e. further into the interior of the wake region.
5 Discussion
In this paper we have shown that two-dimensional Green’s functions
$G_{\unicode[STIX]{x1D6FF}}$
and
$G_{H}$
given by (1.3) and (1.4), respectively, solve the forced Rossby wave equation (1.2). Previously only known via integral representations, these new compact expressions provide a complete description of the response. They reduce on the east–west axis to the long-known exact expressions discussed in § 2, i.e. (2.3a,b
) for
$G_{\unicode[STIX]{x1D6FF}}$
and (2.4a,b
) for
$G_{H}$
and far from the forcing location to the asymptotic forms (4.3) and (4.7), respectively. The latter were also known via transform methods but here they arise directly from the properties of the Bessel functions
$\text{J}_{n}(z),\text{Y}_{n}(z)$
for large (complex) arguments
$z$
.
Exact explicit expressions for Green’s functions were known in one-dimensional (Cahn Reference Cahn1945; Rossby Reference Rossby1945) and three-dimensional (Dickinson Reference Dickinson1969a ,Reference Dickinson b ) settings only. Usually solutions of the forced problem are sought through transform methods (see Veronis Reference Veronis1958; Longuet-Higgins Reference Longuet-Higgins1965; Kamenkovich Reference Kamenkovich1989; Llewellyn-Smith Reference Llewellyn-Smith1997 but also the recent studies by McKenzie Reference McKenzie2014; Webb et al. Reference Webb, Duba and Hu2016). In two-dimensional problems this has led to integral representations like those mentioned in § 2.
A crucial development has been that the integral representation for
$G_{\unicode[STIX]{x1D6FF}}$
can be written as (3.1). From this the closed-form expression (1.3) for
$G_{\unicode[STIX]{x1D6FF}}$
followed via the work of Dixon & Ferrar (Reference Dixon and Ferrar1933) as discussed in § 3. The validity of this solution has been verified in § 3.1 and in § 3.2 we derived and verified the expression (1.4) for
$G_{H}$
. This becomes very efficient through the introduction of the complex conjugate variables
$z_{\star }=\unicode[STIX]{x1D701}+\text{i}\unicode[STIX]{x1D702},\,\bar{z}_{\star }=\unicode[STIX]{x1D701}-\text{i}\unicode[STIX]{x1D702}$
defined in (3.7). In particular, the Rossby wave operator then assumes the symmetric form given in (3.11). That contour patterns of
$G_{\unicode[STIX]{x1D6FF}}$
and
$G_{H}$
shown in § 4 in figures 3 and 5 coincide largely with the parabolas
$\unicode[STIX]{x1D701}=\sqrt{r+x}=\text{const.}$
is perhaps not surprising since if on the right-hand side of (1.2a
) instead of
$\unicode[STIX]{x1D6FF}(t)$
periodic forcing
$\exp (\text{i}\unicode[STIX]{x1D714}t)$
is assumed, lines of constant phase are also
$\unicode[STIX]{x1D701}=\text{const.}$
(see Rhines Reference Rhines, Holton, Curry and Pyle2003).
The parabolic coordinates
$\{\unicode[STIX]{x1D701},\unicode[STIX]{x1D702}\}$
are of wider interest in the context of non-divergent Rossby waves. First note that the form of
${\mathcal{L}}$
in (3.10) suggests that there is no real distinction between
$\unicode[STIX]{x1D701}$
and
$\unicode[STIX]{x1D702}$
whereas in the Cartesian
$\{x,y\}$
formulation longitude
$x$
introduces an ‘asymmetry’ not obvious in (3.10) except for the sign difference between the
$\unicode[STIX]{x1D701}$
and
$\unicode[STIX]{x1D702}$
terms multiplying
$\unicode[STIX]{x1D6FD}$
(the same spatial part of this Rossby wave operator for a steady, frictional version of the vorticity equation was previously considered in ocean circulation context in Maas (Reference Maas1989), Zimmerman & Maas (Reference Zimmerman and Maas1989) and Boyd & Sanjaya (Reference Boyd and Sanjaya2014)). Let us further note that in as much that the Green’s functions are streamfunctions
$\unicode[STIX]{x1D713}$
with the associated velocity components
$u=-\unicode[STIX]{x2202}_{y}\unicode[STIX]{x1D713},v=\unicode[STIX]{x2202}_{x}\unicode[STIX]{x1D713}$
, equation (2.13b
) is simply
$$\begin{eqnarray}\unicode[STIX]{x2202}_{t}(xu+yv)+\frac{1}{2}\unicode[STIX]{x1D6FD}y\,\unicode[STIX]{x1D713}=0,\quad \text{with }xu+vy=\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{r},\quad \boldsymbol{u}=u\boldsymbol{i}+v\boldsymbol{j},\,\boldsymbol{r}=x\boldsymbol{i}+y\boldsymbol{j}\end{eqnarray}$$
and
$\{\boldsymbol{i},\boldsymbol{j}\}$
the customary unit vectors associated with
$\{x,y\}$
. The projection of the velocity vector
$\boldsymbol{u}$
on the position vector
$\boldsymbol{r}$
is
$\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{r}=ru_{r}$
with
$u_{r}$
the radial velocity component in cylindrical coordinates (see figure 1). In other words, in the wave field associated with the Green’s functions the radial velocity component evolves according to
$\unicode[STIX]{x2202}_{t}ru_{r}+(1/2)\unicode[STIX]{x1D6FD}y\unicode[STIX]{x1D713}=0$
. It is not clear to us whether this has a physical meaning. However, leaving out a common factor
$1/2$
, in parabolic coordinates (5.1) becomes
$$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}}\left(\unicode[STIX]{x1D702}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}}-\unicode[STIX]{x1D701}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}}\right)\unicode[STIX]{x1D713}+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D701}\unicode[STIX]{x1D702}\unicode[STIX]{x1D713}=0.\end{eqnarray}$$
Further, differentiation of (5.1) with respect to
$x$
and multiplication of the Rossby wave equation by
$y$
allows for the elimination of the
$\unicode[STIX]{x1D6FD}$
-terms which results in
$$\begin{eqnarray}\unicode[STIX]{x2202}_{t}\{y\unicode[STIX]{x1D6FB}^{2}+2\unicode[STIX]{x2202}_{x}(x\unicode[STIX]{x2202}_{y}-y\unicode[STIX]{x2202}_{x})\}\unicode[STIX]{x1D713}=0\end{eqnarray}$$
but in the parabolic coordinates (5.3) assumes the form
$$\begin{eqnarray}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}}\left\{{\displaystyle \frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}}+{\displaystyle \frac{\unicode[STIX]{x1D702}}{\unicode[STIX]{x1D701}^{2}+\unicode[STIX]{x1D702}^{2}}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}}}+{\displaystyle \frac{\unicode[STIX]{x1D701}}{\unicode[STIX]{x1D701}^{2}+\unicode[STIX]{x1D702}^{2}}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D702}}}\right\}\unicode[STIX]{x1D713}=0.\end{eqnarray}$$
Remarkably, integrating (5.4) in time shows that the Green’s functions are determined by a second-order partial differential equation, which, moreover, is entirely symmetric in parabolic coordinates
$\unicode[STIX]{x1D701},\unicode[STIX]{x1D702}$
. Additionally, the form of the Rossby wave operator
${\mathcal{L}}$
(3.10) in these coordinates suggests that the parabolic coordinates are rather ‘natural’ coordinates for non-divergent Rossby waves.
This raises the question why the response in the forced problem considered here is dominated by the westward parabolas
$\unicode[STIX]{x1D701}=\sqrt{r+x}$
while the opposite parabola
$\unicode[STIX]{x1D702}=\sqrt{r-x}$
only plays a role in providing overall amplitude attenuation via inverse (fractional) powers of
$r=(1/2)(\unicode[STIX]{x1D701}^{2}+\unicode[STIX]{x1D702}^{2})$
. This can be understood with the analyses of Longuet-Higgins (Reference Longuet-Higgins1965) and the more recent work by McKenzie (Reference McKenzie2014), i.e. via considerations of the dispersion relation for monochromatic waves, the group velocity and the method of stationary phase. But a cursory examination of the properties of
${\mathcal{L}}$
in parabolic coordinates, i.e. (3.10), indicates that free, parabolically shaped Rossby waves following lines of constant
$\unicode[STIX]{x1D702}=\sqrt{r-x}$
instead of
$\unicode[STIX]{x1D701}=\sqrt{r+x}$
do also exist at any given frequency. The repercussions of this observation are left for a future study.
A few final remarks are in order: the singular behaviour found for
$G_{\unicode[STIX]{x1D6FF}}$
in (3.15) and
$G_{H}$
in (3.21) at the forcing location persists. For
$G_{H}$
this is not surprising since the forcing is maintained but for
$G_{\unicode[STIX]{x1D6FF}}$
this is counterintuitive since one can interpret
$G_{\unicode[STIX]{x1D6FF}}$
for some
$t=t_{0}>0$
as a given initial streamfunction
$\unicode[STIX]{x1D713}_{0}(x,y)=\unicode[STIX]{x1D713}(x,y,t_{0})$
. This initial condition for
$t=t_{0}$
can be developed in plane Rossby waves with the spectrum
$A_{0}(k,l)$
and for
$t>t_{0}$
the field would evolve as
$$\begin{eqnarray}\unicode[STIX]{x1D713}(x,y,t)=\iint A_{0}(k,l)\text{e}^{\text{i}(\unicode[STIX]{x1D714}t-kx-ly)}\,\text{d}k\,\text{d}l,\end{eqnarray}$$
with
$\unicode[STIX]{x1D714}$
satisfying the well-known dispersion relation
$$\begin{eqnarray}\unicode[STIX]{x1D714}=-{\displaystyle \frac{\unicode[STIX]{x1D6FD}k}{k^{2}+l^{2}}}.\end{eqnarray}$$
Without explicitly knowing the spectrum
$A_{0}(k,l)$
, one would think that dispersion leads to the disappearance of the singularity. But it does not. The ‘shrinking’ of the patterns via the similarity variable
$\unicode[STIX]{x1D6FD}tr$
towards the origin is also baffling although one might see this as a confirmation of the long-known properties of the Rossby waves: short waves (small spatial scales) have small group velocities (Pedlosky Reference Pedlosky1987).
The Rossby wave equation describes the time evolution of vorticity
$q$
which for the Green’s functions is
$q=\unicode[STIX]{x1D6FB}^{2}G$
. This can be calculated quickly. For example, with the variables
$\{z_{\star },\bar{z}_{\star }\},$
introduced in (3.7),
$$\begin{eqnarray}\unicode[STIX]{x1D6FB}^{2}={\displaystyle \frac{4}{z_{\star }\bar{z}_{\star }}}{\displaystyle \frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}z_{\star }\unicode[STIX]{x2202}\bar{z}_{\star }}}\quad \text{and}\quad \quad q_{\unicode[STIX]{x1D6FF}}\equiv \unicode[STIX]{x1D6FB}^{2}G_{\unicode[STIX]{x1D6FF}}={\displaystyle \frac{\unicode[STIX]{x1D6FD}t}{4r}}[\text{J}_{1}(z_{+})\text{Y}_{1}(z_{-})+\text{J}_{1}(z_{-})\text{Y}_{1}(z_{+})]\end{eqnarray}$$
after substitution of (1.3) and using
$\text{J}_{0}^{\prime }=-\text{J}_{1}$
,
$\text{Y}_{0}^{\prime }=-\text{Y}_{1}$
. For large arguments, i.e. far from the forcing, apart from a constant prefactor,
$q_{\unicode[STIX]{x1D6FF}}$
behaves as
$\cos (2a\unicode[STIX]{x1D701})/r^{3/2}$
. The asymptotic energy distribution given in (4.14) behaves as
$\sin ^{2}(2a\unicode[STIX]{x1D701})/r^{2}$
and we thought that perhaps the ‘empty’ spaces seen in the energy distribution in figure 7 might be filled by
$r\times {\mathcal{E}}_{\unicode[STIX]{x1D6FF}}$
with enstrophy
${\mathcal{E}}_{\unicode[STIX]{x1D6FF}}=(1/2)q_{\unicode[STIX]{x1D6FF}}^{2}$
so that a conservation law of the form
$\unicode[STIX]{x2202}_{t}(E_{\unicode[STIX]{x1D6FF}}+r{\mathcal{E}}_{\unicode[STIX]{x1D6FF}})=0$
would follow. This was nearly so, but not exactly. Thus we have no good understanding yet of the
$E_{\unicode[STIX]{x1D6FF}}$
pattern seen in figure 7 nor of the energy distribution
$E_{H}$
in figure 9. A proper energy flux formulation is lacking in terms of recognizable physical quantities (see also Rhines Reference Rhines1975). At best we can draw attention to the salient differences between
$E_{\unicode[STIX]{x1D6FF}}$
and
$E_{H}$
: whereas figures 7 and 8 show that at any distance and given time
$t$
peaks of energy
$E_{\unicode[STIX]{x1D6FF}}$
have equal amplitude and occur both west and east of the forcing, figures 9 and 10 show that the energy
$E_{H}$
associated with the sustained forcing is predominantly found west of the forcing. The peak amplitudes of
$E_{\unicode[STIX]{x1D6FF}}$
do not vary with time
$t$
while with distance
$r$
from the source
$E_{\unicode[STIX]{x1D6FF}}\propto 1/r^{2}$
(see (4.14)) but
$E_{H}$
grows with time
$t$
and decays more rapidly, i.e.
$E_{H}\propto 1/r^{3}$
(see (4.19)). The temporal behaviour is not surprising in that continued forcing leads to growth in energy but the spatially more confined nature of
$E_{H}$
as compared to that of
$E_{\unicode[STIX]{x1D6FF}}$
is surprising.
In as much in that the barotropic, non-divergent Rossby wave is a small but fundamental element in the realm of theoretical geophysical fluid dynamics, this study may stimulate interest beyond the current study to search for simple closed-form solutions of forced divergent Rossby waves (waves with finite Rossby deformation radius). This introduces finite group velocity for very long waves and more can then perhaps be said about the evolution of energy distributions and may help recognize ‘where’ wave energy goes. In view of our results there is reason to be optimistic that the complicated integral representation of Veronis (Reference Veronis1958) for divergent Rossby waves and the elegant result of Webb et al. (Reference Webb, Duba and Hu2016) can be reduced to simpler expressions.
Another possible strategy is to consider a ‘switch-on/switch-off’ source, that is, the response
$G_{\unicode[STIX]{x1D6FF}}$
which Veronis (Reference Veronis1958) called a ‘tweak’, followed by an ‘anti-tweak’. This will eliminate the singularity at the forcing location and energy will subsequently be finite everywhere. The simplicity of our expression for
$G_{\unicode[STIX]{x1D6FF}}$
(1.3) allows for quick visualization of such a scenario and further mathematical analysis but this too is left for future research.
Acknowledgements
R.C.K. acknowledges support from National Science Foundation grants OCE 10-32256 and OCE 15-38559. We thank two anonymous referees and S. Llewellyn Smith for their insightful comments of the manuscript.
