2. Linear Rossby wave theory

The simplest description of a planetary Rossby wave on a beta plane for a rigid-lid, homogeneous-density ocean or atmosphere is in linear approximation given by

(2.1)\begin{equation} {\partial}_t q + \beta v = \textrm{curl}\, \boldsymbol{\tau}. \end{equation}

Throughout this paper we set density $\rho =1$ for convenience. Here, relative vorticity $q= {\partial }_x v- {\partial }_y u$ is created by south-north advection of planetary vorticity and by $\boldsymbol {\tau }=\{\tau _x,\tau _y\}$ the wind-stress vector, see e.g. Veronis (Reference Veronis1958) and Pedlosky (Reference Pedlosky1987). As usual, $t$ is time, $x$ longitude, $y$ latitude and $\beta v$ is the advection of planetary vorticity by the latitudinal (south-north) velocity component $v$, where $\beta$ represents the magnitude of the northward directed planetary vorticity gradient.

Incompressibility, $\boldsymbol {\nabla } \boldsymbol {\cdot } {\boldsymbol {u}}=0$, allows introducing a streamfunction $\psi$, so that the velocity components are $u=- {\partial }_y\psi , v= {\partial }_x \psi$ and $q=\nabla ^2 \psi$, with $\nabla ^2=( {\partial }_x^2+ {\partial }_y^2)$. ‘Free’ Rossby waves are governed by (2.1) with $\textrm {curl}\,\boldsymbol {\tau }=0$ on the right-hand side:

(2.2)\begin{equation} {\partial}_t \nabla^2 \psi+ \beta {\partial}_x\psi = 0. \end{equation}

Multiplication of (2.2) by $\psi$ yields a conservation law for kinetic energy

(2.3)\begin{equation} T=\tfrac{1}{2}\boldsymbol{\nabla}\psi\boldsymbol{\cdot}\boldsymbol{\nabla}\psi, \end{equation}

that reads

(2.4)\begin{equation} \frac{{\partial} T}{{\partial} t} +\boldsymbol{\nabla} \boldsymbol{\cdot}{\boldsymbol{F}}_T=0\quad \text{with}\ {\boldsymbol{F}}_T={-}\psi \frac{{\partial} \boldsymbol{\nabla} \psi}{{\partial} t}- {\boldsymbol{i}} {{\frac{1}{2}}} \beta \psi^2, \end{equation}

where we use ${\boldsymbol {i}, \boldsymbol {j}}$ to denote the customary unit vectors associated with Cartesian $x,y$-directions, respectively. Buchwald (Reference Buchwald1972) established by inspection of (2.2) that the Rossby wave equation gives rise to a second conservation law, related to

(2.5)\begin{equation} V=\tfrac{1}{2}\beta \psi\eta \end{equation}

which acts as the density function of pseudo-potential energy per unit mass, despite the presence of a rigid upper surface. Here, $\eta$ denotes the northward particle displacement whose time derivative yields the northward velocity $v= {\partial }\eta / {\partial } t$.

Without use of Buchwald's results, the conservation equation for pseudo-potential energy $V$ can be obtained also by using the observation by Seliger & Whitham (Reference Seliger and Whitham1968) that the linear non-divergent Rossby wave equation (2.2) follows with a variational principle akin to Hamilton's principle in classical mechanics: the dynamics is determined by requiring the difference of kinetic energy and potential energy to be stationary with respect to small variations.

By introducing an auxiliary function $\chi$, such that $\psi = {\partial }_t \chi$, Seliger & Whitham (Reference Seliger and Whitham1968) note that (2.2) is obtained by demanding that the variation $\delta \textrm {I}$ of an action integral $\textrm {I}=\iiint L\, \textrm {d} x \,\textrm {d} y \,\textrm {d} t$ with respect to small variations $\delta \chi$ vanishes. Here, $L$ is the Lagrangian density

(2.6)\begin{equation} L=T-V={{\frac{1}{2}}}\left(\frac{{\partial} \boldsymbol{\nabla} \chi}{{\partial} t}\right)^2 -{{\frac{1}{2}}} \beta \frac{{\partial} \chi}{{\partial} t}\frac{{\partial} \chi}{{\partial} x}. \end{equation}

With the definition of $\chi$ we have $\psi = {\partial }_t\chi$ but also $ {\partial }_x\chi =\eta$ because northward velocity $v= {\partial }_t \eta = {\partial }_x \psi = {\partial }^2_{x,t}\chi$. Thus $V$ in (2.6) equals the pseudo-potential energy (2.5) of Buchwald (Reference Buchwald1972). As Olbers, Willebrand & Eden (Reference Olbers, Willebrand and Eden2012) point out, a pseudo-potential energy appearing in the expression for the Lagrangian density is not necessarily identical to a physical energy, although the interpretation of (pseudo) kinetic energy $T$ as a physical energy is beyond dispute.

Using the Rossby wave equation (2.2), $V$ in (2.5) can be written elegantly as

(2.7)\begin{equation} V={-}{{\tfrac{1}{2}}} \psi \nabla^2 \psi ={-}{{\tfrac{1}{2}}} \psi q \quad \text{with}\ q=\nabla^2\psi. \end{equation}

Definitions (2.3) and (2.7) show that the Lagrangian density can be written as the divergence of a flux:

(2.8)\begin{equation} L\equiv T-V={{\tfrac{1}{2}}} \boldsymbol{\nabla} \boldsymbol{\cdot} (\psi\boldsymbol{\nabla} \psi). \end{equation}

Subtracting the time derivative of $T-V$ in (2.8) from (2.4) yields the conservation law for pseudo-potential energy $V$:

(2.9)\begin{equation} \frac{{\partial} V}{{\partial} t} +\boldsymbol{\nabla} \boldsymbol{\cdot}{\boldsymbol{F}}_V=0 \quad \text{with}\ {\boldsymbol{F}}_V={{\frac{1}{2}}} \frac{{\partial} \psi}{{\partial} t} \boldsymbol{\nabla} \psi-{{\frac{1}{2}}} \psi \boldsymbol{\nabla} \frac{{\partial} \psi}{{\partial} t}-{\boldsymbol{i}} {{\frac{1}{2}}} \beta \psi^2. \end{equation}

Since $E=T+V$, combining (2.4) and (2.9) implies the energy conservation law

(2.10)\begin{equation} \frac{{\partial} E}{{\partial} t} +\boldsymbol{\nabla}\boldsymbol{\cdot} {\boldsymbol{F}}=0 \quad \text{with} \ {\boldsymbol{F}}={\boldsymbol{F}}_T + {\boldsymbol{F}}_V={-}{{\frac{3}{2}}}\psi \frac{{\partial} \boldsymbol{\nabla} \psi}{{\partial} t}+{{\frac{1}{2}}} \frac{{\partial} \psi}{{\partial} t} \boldsymbol{\nabla} \psi- {\boldsymbol{i}} \beta \psi^2. \end{equation}

The relevance of acknowledging $V$ as a form of pseudo-potential energy in (2.7) is immediately clear if we apply it to a free monochromatic Rossby wave: $\psi =A \sin \varPhi$ with amplitude $A$ and phase $\varPhi \equiv kx+ly- \omega t- \phi$, where wave vector ${\boldsymbol {k}}=(k,l)$ obeys the dispersion relation

(2.11)\begin{equation} \omega ={-} \beta \frac{k}{\kappa^2}, \quad \kappa=|{\boldsymbol{k}}|,\quad \omega > 0. \end{equation}

Both kinetic and pseudo-potential energy,

(2.12a,b)\begin{equation} T= {{\tfrac{1}{2}}} \kappa^2 A^2 \cos^2 \varPhi \quad \text{and}\quad V= {{\tfrac{1}{2}}} \kappa^2 A^2 \sin^2 \varPhi, \end{equation}

propagate on average with the group velocity ${\boldsymbol {c}}_g=( {\partial } \omega / {\partial } k, {\partial } \omega / {\partial } l)$. These expressions do not only imply, as Buchwald inferred, equipartitioning of their phase-averages, where the phase-average is defined as $\langle \cdot \rangle = (1/2{\rm \pi} )\int _0^{2{\rm \pi} } \cdot \textrm {d}\varPhi ,$ but they also render the total pseudo-energy

(2.13)\begin{equation} E= T+V={{\tfrac{1}{2}}} \kappa^2 A^2 \end{equation}

a spatially uniform constant, without the need for any phase-averaging. Remarkably, an infinite-dimensional system thus seems to recover a local balance between kinetic and pseudo-potential energy, a property otherwise restricted to single degree of freedom systems, such as the pendulum.

For plane non-divergent Rossby waves, the group velocity vector is given by

(2.14)\begin{equation} {\boldsymbol{c}}_g=\left(\frac{{\partial} \omega}{{\partial} k},\frac{{\partial} \omega}{{\partial} l}\right)={\boldsymbol{i}}\frac{\beta(k^2-l^2)}{\kappa^4} +{\boldsymbol{j}}\frac{2\beta kl}{\kappa^4}. \end{equation}

The energy flux vector

(2.15)\begin{align} {\boldsymbol{F}}&= \frac{\beta A^2}{2\kappa^2}\left({\boldsymbol{i}}\left[k^2-l^2+ l^2\cos(2\varPhi)\right]+{\boldsymbol{j}}\,lk\left[2-\cos(2\varPhi)\right]\right)\nonumber\\ &={\boldsymbol{c}_g}E + {{\frac{1}{2}}} \beta A^2\frac{l }{\kappa^2}\cos(2\varPhi) {\boldsymbol{k}^{{\perp}}}, \end{align}

where ${\boldsymbol {k}^{\perp }}=(l,-k)$ is a vector perpendicular to ${\boldsymbol {k}}$. This shows that, as usual in wave dynamics, the phase-average of the flux vector

(2.16)\begin{equation} \langle {\boldsymbol{F}}\rangle={\boldsymbol{c}}_g E \end{equation}

is indeed the product of group velocity (2.14) and energy density (2.13). This spatially constant vector is obviously non-divergent. Yet it shows the presence of a steady flux of energy into the direction into which the group velocity points. The second, oscillatory part of the flux vector proportional to $\cos (2 \varPhi )$, that accommodates the exchange of kinetic to pseudo-potential energy and back, consists of a vector that is perpendicular to wave vector ${\boldsymbol {k}}$, implying this part is indeed also non-divergent. Vanishing of the divergence of energy flux ${\boldsymbol {F}}$ in (2.15) agrees with the fact that pseudo-energy density $E$ is invariant over time. In § 4, we show that this also holds for the much more involved response to an impulsive point source of vorticity which we shall first describe briefly in the next section.

3. Rossby waves due to an impulsive point source: Green's function

With (2.12a,b) we saw that for a plane monochromatic Rossby wave, the kinetic energy $T$ and pseudo-potential energy $V$ complement each other perfectly: where $T$ is high, $V$ is low and vice versa, in such a manner that their total sum $E$ is spatially uniform. In § 4, we show that this also holds for the wave field generated by an instantaneous application of wind-stress torque, $\textrm {curl}\, \boldsymbol {\tau }$, infinitely concentrated at a singular point. This complicated Rossby wave field generated by an impulsive point source, the Green's function $G$, is described in detail in Kloosterziel & Maas (Reference Kloosterziel and Maas2017). This Green's function is the solution of

(3.1)\begin{equation} {\mathcal{L}} G = \delta(t) \delta(x)\delta(y) \quad \text{with}\ {\mathcal{L}} = \frac{{\partial}}{{\partial} t}\nabla^2 + \beta \frac{{\partial}}{{\partial} x}. \end{equation}

Here ${\mathcal {L}}$ is the Rossby wave operator and $\delta$ the usual Dirac delta-function. In Kloosterziel & Maas (Reference Kloosterziel and Maas2017), we showed that

(3.2)\begin{equation} G = \frac{H(t)}{4} [\textrm{J}_0(z_+)\textrm{Y}_0(z_-)+\textrm{J}_0(z_-)\textrm{Y}_0(z_+)], \end{equation}

where $\textrm {J}_n, \textrm {Y}_n$ are $n$th-order Bessel functions of the first and second kind, respectively, with complex conjugate arguments

(3.3)\begin{equation} z_\pm{=}\sqrt{\beta t (x \pm \textrm{i} y)}=\sqrt{\beta t r}\textrm{e}^{{\pm} \textrm{i} \theta/2}, \end{equation}

with $r, \theta$ the usual cylindrical coordinates in the $x,y$ plane. The complex conjugate variables $z_\pm$ can be expressed also in parabolic coordinates $\{\zeta , \eta \}$

(3.4a–e)\begin{equation} z_+{=}az_\star, \quad z_-{=}a\bar{z}_\star, \quad \quad a=\sqrt{\frac{\beta t}{2}}, \quad z_\star{\equiv}\zeta + {\textrm{i} }\eta, \quad \bar{z}_\star{\equiv}\zeta -{\textrm{i} }\eta \end{equation}

with

(3.5)\begin{equation} \zeta= \sqrt{x+r},\quad \eta= \sqrt{r-x} \quad \textrm{with} \ r=\sqrt{x^2+y^2}, \quad \zeta^2+\eta^2=2r. \end{equation}

The Green's function has a self-similar dependence on $\beta t r$, and an independent angular dependence on $\theta$. The prefactor $H(t)$ in (3.2) is the Heaviside unit stepfunction which enforces ‘causality’, i.e. $G=0$ for $t< 0$. By definition $\delta (t)\equiv \textrm {d} H(t)/\textrm {d} t$. In the following, $H(t)$ will be omitted with the understanding that all results are for $t>0$. The Green's function is remarkable in that its closed-form expression requires the use of products of Bessel functions of complex arguments, even though the solution itself is real. Graphing $G$ given in (3.2) at various times is easy with available numerical packages, such as Wolfram Research, Inc. (2020).

Figure 1 displays the Green's function during its development. Initially, it consists of an isotropic response as the highest derivative term in the differential operator $\mathcal {L}$ in (3.1), representing the evolution of relative vorticity, which should match the point-source singularity. Hence, as on a non-rotating plane ($\beta =0$), the fluid initially responds in the form of a circular irrotational point or line vortex. In figure 1(a), the near circular symmetry near the origin at the start is obvious. Mathematically, this initial behaviour and the behaviour near the origin follows with the small-$z$ behaviour of $\textrm {J}_0,\textrm {Y}_0$ in (3.2) (see (A1) in appendix A):

(3.6)\begin{equation} \lim t r\downarrow 0: \quad G \sim \frac{1}{2 {\rm \pi}} \ln r \quad \text{and swirl velocity}\quad v_\theta= \frac{{\partial} G}{{\partial} r} \sim \frac{1}{2{\rm \pi} r}. \end{equation}

The point vortex’s localized vorticity is positive, in agreement with the cyclonic nature of the applied curl of the wind stress. Outside the origin, the response initially consists of potential flow, lacking vorticity. Thus, initially all energy is kinetic, i.e. $V=0$, and with (3.6)

(3.7)\begin{equation} \lim t\downarrow 0: E=T=\frac{1}{2}v_\theta^2=\frac{1}{2}\left(\frac{1}{2{\rm \pi} r}\right)^2. \end{equation}

However, soon after, relative vorticity $q$ is created by cyclonic (counter-clockwise) advection of northwards-increasing background vorticity ($\beta v$ in (2.1) or the second term of operator $\mathcal {L}$ in (3.1)). The exact expression for the vorticity $q=\nabla ^2G$ is given below in (4.3a,b). In figure 2, we show graphs of $\nabla ^2 G$ at the same times for which $G$ is shown in figure 1. Due to this advection, the secondary vortices acquire positive sign to the west, and negative sign to the east of the source. This is seen in the first two panels of figure 2. At the central latitude $y=0$, these secondary vorticity fields in turn advect background vorticity, in both cases leading to positive tertiary vortices further to the west and to the east of the secondary vortices, respectively (see figure 2c). This process continues, to the east leading to an array of vortices alternating in sign, while those to the west are all positive. This leads to a single-signed, algebraically decreasing tail of vorticity to the west (proportional to $r^{-3/2}$) while to the east a set of westward propagating vortices appear.

Figure 2.

Vorticity of the Green's function, $q=\nabla ^2G$ (4.3a,b) at several moments in time $t$ over a fixed spatial area. Vorticity increases with time. To enable use of the same colour scale, $q$ is scaled by $\sqrt {t}$ and has been multiplied by $10^3$. Parabolas and times as in figure 1: (a) $t=0.005$, (b) $t=0.05$ and (c) $t=0.5$.

The patterns seen in figure 1(c) for $G$ and figure 2(c) for $q=\nabla ^2G$ at time $t=0.5$ can be described accurately by simple analytical expressions found with expansions of the Bessel function for large argument $z$ (see (A3) in appendix A). For example, for large $\beta tr$

(3.8)\begin{align} G=\frac{1}{4}[\textrm{J}_0(z_+)\textrm{Y}_0(z_-)+\textrm{J}_0(z_-)\textrm{Y}_0(z_+)] \sim{-}\frac{\cos(z_+{+}z_-)}{2{\rm \pi} \sqrt{z_+z_-}}={-}\frac{\cos(\sqrt{2\beta t(x+r)})}{2{\rm \pi} \sqrt{\beta tr}}. \end{align}

Lines of constant $\zeta =\sqrt {x+r}$ are parabolas (see figure 1) that open towards the west ($x<0$). A number of these parabolas have been plotted in figure 1. Veronis (Reference Veronis1958) already predicted that lines of constant phase would be such parabolas. For a more detailed description of $G$, see Kloosterziel & Maas (Reference Kloosterziel and Maas2017).

With the advection of planetary vorticity a transfer of energy from $T$ to $V$ is enabled by means of Rossby waves, making it likely that also the total pseudo-energy, $E=T+V$, evolves. In the next section we show however that $E$ remains invariant as well as isotropic, i.e. at all times $E$ equals the initial kinetic energy (3.7).

4. Energy for the Green's function wave field

In the following, it will be expedient to use the following notation:

(4.1)\begin{equation} \textrm{I}_{mn}\equiv \textrm{J}_m(z_+)\textrm{Y}_n(z_-)+\textrm{J}_n(z_-)\textrm{Y}_m(z_+), \end{equation}

where $m,n$ are integers. For brevity, we suppress the spatio-temporal dependence on the complex $z_{\pm }$ coordinates defined in (3.3). With this notation the Green's function (3.2) is

(4.2)\begin{equation} G = \frac{1}{4} \textrm{I}_{00}. \end{equation}

Treating $G$ as the streamfunction $\psi$ of the forced problem ${\mathcal {L}}\psi =\delta (t)\delta (x)\delta (y)$, using (3.4a–e) it is easy to calculate the vorticity $q\equiv \nabla ^2\psi =\nabla ^2G$:

(4.3a,b)\begin{equation} {\nabla^2=\frac{4}{z_\star\bar{z}_\star}\frac{{\partial}^2}{{\partial} z_\star{\partial} \bar{z}_\star}} \quad \text{and} \quad q=\nabla^2 G= \frac{\beta t }{2(\zeta^2+\eta^2)}\textrm{I}_{11} \end{equation}

because $\{\textrm {J}_0',\textrm {Y}_0'\}=-\{\textrm {J}_1,\textrm {Y}_1\}$ (primes indicate differentiation). The variables $z_\star ,\bar {z}_\star ,\zeta ,\eta$ are defined in (3.4a–e) and (3.5). Figure 2 shows this vorticity field at three instances. In Kloosterziel & Maas (Reference Kloosterziel and Maas2017), we already determined that the kinetic energy $T=(\boldsymbol {\nabla } \psi \boldsymbol {\cdot }\boldsymbol {\nabla } \psi )/2$ associated with the streamfunction $\psi =G$ equals

(4.4)\begin{equation} T = \frac{1}{2(\zeta^2+\eta^2)} \left[\left(\frac{{\partial} G}{{\partial} \zeta}\right)^2 +\left(\frac{{\partial} G}{{\partial} \eta}\right)^2\right] =\frac{1}{4^2} \frac{\beta t}{\zeta^2+\eta^2} \textrm{I}_{01}\textrm{I}_{10}. \end{equation}

The evolution of the kinetic energy distribution $T$ is shown in figure 3. Late in its development, the kinetic energy's spatial distribution shows regions of very small values, see light yellow areas in figure 3(c). This suggests these must be regions where energy is represented by an appropriately defined form of potential energy. Substituting $\psi =G$ from (3.2) and $\nabla ^2\psi =\nabla ^2G$ from (4.3a,b) in (2.7) we find

(4.5)\begin{equation} V={-}{{\frac{1}{2}}} \psi \nabla^2\psi ={-}{{\frac{1}{2}}} G\nabla^2G ={-} \frac{1}{4^2}\frac{\beta t}{\zeta^2+\eta^2}\textrm{I}_{00}\textrm{I}_{11}. \end{equation}

The pseudo-potential energy $V$ is shown in figure 4 for the same three times used in figure 3 for the kinetic energy $T$. Comparison of figure 4(c) with figure 3(c) shows that $V$ indeed complements $T$ in that where $T$ has low values, $V$ has high values and vice versa.

Figure 3.

Kinetic energy of the Green's function, $T=(\boldsymbol {\nabla } G \boldsymbol {\cdot } \boldsymbol {\nabla } G)/2$ (4.4). Times as in previous figures 1 and 2: (a) $t=0.005$, (b) $t=0.05$ and (c) $t=0.5$. Parabolas also as in figures 1 and 2.

Figure 4.

Pseudo-potential energy of the Green's function $V=-(G \nabla ^2G)/2$ (4.5) at times (a) $t=0.005$, (b) $t=0.05$ and (c) $t=0.5$ over a fixed spatial area. Parabolas and times as in figures 1, 2 and 3. Colour scale as in figure 3. Final panel (d) shows the total energy $E=T+V$ (4.8), which is time-invariant and isotropic (see text).

For reference, the same parabolas as in figure 1 have been shown as dashed lines in figures 3(b,c) and 4(b,c), passing through the maxima of $T$ and the minima of $V$. Note that the pseudo-potential energy can become negative, initially mimicking the vorticity's sign (compare first panels of figures 2 and 4) because initially $-G>0$ everywhere, while, later, weak negative values of $V$ contract to regions where $T$ is maximal.

When (4.4) and (4.5) are added, four terms involving the product of four Bessel functions cancel and the result is

(4.6)\begin{align} &E=T+V= \frac{1}{4^2} \frac{\beta t}{\zeta^2+\eta^2}\nonumber\\&\quad \times \left[ \textrm{J}_0(z_+)\textrm{Y}_1(z_+)-\textrm{J}_1(z_+)\textrm{Y}_0(z_+)\right] \left[ \textrm{J}_0(z_-)\textrm{Y}_1(z_-)-\textrm{J}_1(z_-)\textrm{Y}_0(z_-)\right]. \end{align}

This is a product of two Wronskians $\mathcal {W}$ (see the NIST Handbook of Mathematical functions Olver et al. (Reference Olver, Lozier, Boisvert and Clark2010, formula 10.5.2)):

(4.7)\begin{equation} \textrm{J}_0(z)\textrm{Y}_1(z)-\textrm{J}_1(z)\textrm{Y}_0(z)=\mathcal{W}\{\textrm{J}_0,\textrm{Y}_0\} =\frac{2}{{\rm \pi} z}\quad \text{and because}\ \frac{\beta t}{\zeta^2+\eta^2}=\frac{z_+z_-}{2 r^2} \end{equation}

(see definitions ((3.4a–e)–(3.5))), we find the astonishing result that at all times

(4.8)\begin{equation} E = T+V=\frac{1}{2}\left(\frac{1}{2{\rm \pi} r}\right)^2. \end{equation}

This is displayed in figure 4(d). Hence, the total energy is isotropic and time-independent ($ {\partial } E/ {\partial } t=0$). Note this is the energy entirely contained initially in the kinetic energy of the line vortex generated by the impulsive forcing, see (3.7). At later times, kinetic energy $T$ is transferred to pseudo-potential energy $V$. This is illustrated in figure 5, which shows the angular dependence of $T/E$ and $V/E$ on a circle of radius $r=40$. Due to the self-similar development of the streamfunction field, the particular radius chosen to show its angular distribution is arbitrary. The temporal development of rapid undulations in the angular distributions of $T$ and $V$ seen in figure 5 are due to the appearance of Rossby waves of smaller and smaller scales.

Figure 5.

Angular distribution of normalized kinetic energy $T/E$ (solid) and pseudo-potential energy $V/E$ (dashed), scaled by total energy $E$ along circle of radius $r=40$ at times (a) $t=0.0005$, (b) $t=0.01$ and (c) $t=0.2$ (time increasing by a factor 20). While all energy $E$ starts out kinetic energy $T$, it rapidly converts into potential energy $V$ along the west axis ($\theta ={\rm \pi}$, centre of graphs), but at all times $T/E+V/E=1$ (thin solid line).

How the isotropy and time-independence of $E$ come about is seen with the asymptotic properties of $T$ and $V$. They are determined by the large-$z$ behaviour of the Bessel functions in $\textrm {I}_{mn}$ and we find that for large $\beta tr$ (see appendix A):

(4.9a)\begin{align} \beta t r\rightarrow \infty: &\quad T\sim \frac{1}{2} \frac{\sin^2(2a\zeta)}{{\rm \pi}^2(\zeta^2+\eta^2)^2}= \frac{1}{2}\frac{\sin^2(\sqrt{2\beta t(x+r)})}{(2{\rm \pi} r)^2}, \end{align}

(4.9b)\begin{align} &\quad V\sim \frac{1}{2} \frac{\cos^2(2a\zeta)}{{\rm \pi}^2(\zeta^2+\eta^2)^2} = \frac{1}{2}\frac{\cos^2(\sqrt{2\beta t(x+r)})}{(2{\rm \pi} r)^2}. \end{align}

Because $\sin ^2 z +\cos ^2 z=1$, these approximations also yield the exact result (4.8) for the total pseudo-energy $E=T+V$. The patterns for both $T$ and $V$ have a distinct ‘preference’ for the westward side of the forcing – at least visually (see figures 3 and 4) – but when added to form $E$ this disappears and in accordance with (4.8) isotropy results. Note that this is analogous to the result for plane waves, discussed in § 2, leading to constant total energy everywhere.

Kinetic energy may locally increase beyond that given by the isotropic distribution present after the instantaneous torque has been applied ($T/E>1$, see first two panels of figure 5). At the same time, pseudo-potential energy $V$ may initially become negative. This is a consequence of the material conservation of the combined planetary and relative vorticity. Advection of an initially quiescent fluid that is endowed with a meridionally varying planetary vorticity, leads to the generation of relative vorticity. Its motions enhance or diminish $T$ locally, depending on the increase or decrease of associated gradients in $G$. This is possible because the beta plane's quiescent ground state $({\boldsymbol {u}}=0)$ is described from within a non-uniformly rotating frame of reference, a frame that carries non-zero ground-state energy (Tolstoy Reference Tolstoy1973). For stable wave motions, this energy may temporarily raise the kinetic energy above its initially injected amount, at the expense of $V$ turning negative. In that sense, $V$ denotes a surplus $(V>0)$ or shortage $(V<0)$ of background energy. This happens evidently without affecting that background state owing to the huge velocity difference between wave velocity and speed with which the beta plane revolves around Earth's axis. The rotating system represents an effectively infinite reservoir of energy. The presence of such a fund of energy is more easily recognized in a stable shear-flow, $U$, lacking inflection points and sheared either in the horizontal or vertical direction. Shear-flow stability waves, supported by curvature in this background flow, $U'' \neq 0$, are isomorphic with Rossby waves on the beta plane (Tolstoy Reference Tolstoy1973; Harnik & Heifetz Reference Harnik and Heifetz2007).

5. Radial energy flux for the Green's function wave field

In view of (2.10), when the pseudo-energy is invariant, $ {\partial } E/ {\partial } t=0$, the energy flux ${\boldsymbol {F}}$ must be non-divergent:

(5.1)\begin{equation} \boldsymbol{\nabla} \boldsymbol{\cdot} {\boldsymbol{F}}=0, \end{equation}

everywhere outside the origin. Non-divergency of the energy flux, however, does not imply that the energy flux itself should vanish. In the presence of an initial amount of energy, introduced when the wind-stress torque is applied at $t=0$, a certain amount may escape, and this is indeed what we will find.

Using our solution for the Green's function, we can plot the energy flux vector ${\boldsymbol {F}}$ given in (2.10) over the plane, or, as we do here in figure 6, present it upon division by the invariant energy $E(r)$ given in (4.8). We can interpret this as the energy velocity vector ${\boldsymbol {U}} \equiv {\boldsymbol {F}}/E$ that can be used to construct energy streamlines and energy paths (Chapman Reference Chapman2001). Since the energy flux satisfies the two-dimensional continuity equation (5.1), it can be described by an energy streamfunction whose streamlines are tangent to the velocity vectors depicted in figure 6.

Figure 6.

Energy velocity, defined as the energy flux scaled by total energy, ${\boldsymbol {U}} = {\boldsymbol {F}}/E$: magnitude (colour) and direction (arrows) at times (a) $t=0.1$, (b) $t=0.4$ and (c) $t=1$.

Figure 6 shows that energy is carried in a non-uniform manner strictly outwards, away from the source. To show this analytically, we determined the outward-directed energy flux across a circle of some fixed radius $r$ about the source-origin as a function of the azimuthal angle $\theta$. In polar coordinates, the energy flux ${\boldsymbol {F}}=F_r{\boldsymbol {e}}_r+F_\theta {\boldsymbol {e}}_\theta$. Here, ${\boldsymbol {e}}_r, {\boldsymbol {e}}_\theta$ denote the orthogonal unit vectors associated with the polar coordinate system. Of interest is only the radial component, $F_r$, which follows with the fact that $\boldsymbol {\nabla }={\boldsymbol {e}}_r {\partial }/ {\partial } r + {\boldsymbol {e}}_\theta (1/r) {\partial }/ {\partial } \theta$. Thus, with (2.10),

(5.2)\begin{equation} F_r={-}{{\frac{3}{2}}} \psi \frac{{\partial}^2 \psi}{{\partial} t{\partial} r}+ {{\frac{1}{2}}} \frac{{\partial} \psi}{{\partial} t} \frac{{\partial} \psi}{{\partial} r} - \beta \psi^2\cos \theta \end{equation}

because ${\boldsymbol {e}}_r \boldsymbol {\cdot } {\boldsymbol {i}}=\cos \theta$.

Since $\psi =G=\textrm {I}_{00}/4$ is a function of just $z_\pm$ (see (4.1)), the derivatives $ {\partial }_t$ and $ {\partial }_r$ in (5.2) are quickly calculated with the fact that according to (3.3) $ {\partial } z_\pm / {\partial } t =z_\pm /2t, \, {\partial } z_\pm / {\partial } r=z_\pm /2r$. Further, the second derivative $ {\partial }^2\psi / {\partial } t {\partial } r$ can be simplified by noting that $\textrm {J}_0, \textrm {Y}_0$ satisfy their defining Bessel equation

(5.3)\begin{equation} \left(z{{\partial}\over {\partial} z}\right)^2\{\textrm{J}_0,\textrm{Y}_0\}={-}z^2\{\textrm{J}_0,\textrm{Y}_0\}. \end{equation}

Also taking into account that $\{\textrm {J}_0',\textrm {Y}_0'\}=-\{\textrm {J}_1,\textrm {Y}_1\}$ and that $z_+^2+z_-^2=2\beta t r\cos \theta$, we find

(5.4)\begin{equation} F_r ={-} \frac{1}{8}\frac{1}{ 4^2}\frac{1}{tr}\left\{(z_+^2+z_-^2)\textrm{I}_{00}^2 +6z_+z_- \textrm{I}_{00}\textrm{I}_{11}- [z_+ \textrm{I}_{10} +z_- \textrm{I}_{01}]^2\right\}, \end{equation}

with the $\textrm {I}_{mn}$ defined in (4.1). This exact result for the radial flux is shown in figure 7 as a function of $\theta$ at three instants and fixed radius $r=1$. It is seen in figure 7 that the radial component of the energy flux vector ${\boldsymbol {F}}$ associated with $E$ is always directed outwards, away from the source. In contrast, we find that the kinetic energy flux ${\boldsymbol {F}}_T$ for the traditional Rossby wave energy equation (2.4) exhibits inward-directed fluxes over some range of polar angles (not shown). The energy flux, ${\boldsymbol {F}}_V$, present in the conservation equation for $V$, (2.9), however, does exhibit only an outward-directed flux. This makes sense as initially all vorticity is concentrated at the source.

Figure 7.

Radial energy flux $F_r$ through the unit circle ($r=1$) at times $t=1/3,1,3$ as a function of azimuthal coordinate $\theta$ given by its exact expression (5.4) or (5.8) (solid) and, for $t=1$, by its asymptotic approximation (5.6) (dashed). The centre of this graph $\theta ={\rm \pi}$ corresponds to west. Horizontal lines give the isotropic part $1/(8 {\rm \pi}^2 tr)$ of $F_r$ at the corresponding moments.

The complicated expression (5.4) for the radial flux can be well-approximated in the limits of small and large $\beta tr$. Substitution in (5.4) of the small-$z$ expansions of the Bessel functions from (A1) in appendix A yields

(5.5)\begin{equation} \beta t r \rightarrow 0: F_r\sim \frac{1}{8{\rm \pi}^2 tr} \left[1-\frac{1}{2}\beta tr \cos\theta \log^2(\beta t r) +O\left((\beta tr)^2\log^2(\beta t r)\right)\right], \end{equation}

while with substitution of the large-$z$ expansions from (A3) we find after rearrangements and use of trigonometric identities with (A5) in appendix A that

(5.6)\begin{equation} \beta t r \rightarrow \infty: F_r\sim \frac{1}{8{\rm \pi}^2 tr}\left [1+\sin^2 \left({{\frac{1}{2}}} \theta\right)\cos\left({ 4}\sqrt{\beta t r}\cos\left({{\frac{1}{2}}} \theta\right)\right)\right]. \end{equation}

This is positive for all $\theta$ and shown as dashed line in figure 7 for a single time $t=1$ at $r=1$. On the central latitude ($y=0$)

(5.7a,b)\begin{equation} \beta t r \rightarrow \infty: \text{`east'} (\theta=0, 2{\rm \pi}): F_r\sim \frac{1}{8{\rm \pi}^2 tr}, \quad \text{`west'} (\theta= {\rm \pi}): F_r\sim \frac{1}{4 {\rm \pi}^2 tr}. \end{equation}

In this limit along the central latitude, twice as much energy propagates to the ‘west’. For small $\beta tr$, (5.5) shows that there is also a (slight) preference for westward energy propagation.

With a non-trivial analysis, we show in appendix A that when $\psi =G$ the radial flux consists of an isotropic part ($\theta$-independent) and an anisotropic part $\mathcal {F}(r,\theta ,t)$:

(5.8)\begin{equation} F_r=\frac{1}{8{\rm \pi}^2 tr}+\mathcal{F}(r,\theta,t) \quad \text{with} \ \mathcal{F} ={-}\frac{2}{8}\frac{1}{4^2}\frac{1}{ tr}\frac{{\partial}^2}{{\partial} \theta^2} \textrm{I}_{00}^2={-} \frac{1}{4 tr}\frac{{\partial}^2}{{\partial} \theta^2} G^2. \end{equation}

The isotropic part of the outward-directed energy flux $1/(8{\rm \pi} ^2 tr)$ is recognized in both asymptotic expressions (5.5) and (5.6).

Because of continuity of the (squared) Green's function and its azimuthal derivatives,

(5.9)\begin{equation} \oint F_r r \,\textrm{d} \theta =\oint \left(\frac{1}{8{\rm \pi}^2 tr}+\mathcal{F}\right) r \,\textrm{d}\theta =\oint \frac{1}{8{\rm \pi}^2 tr}r\,\textrm{d}\theta= \frac{1}{4{\rm \pi} t} \end{equation}

for any radius $r$. Thus, the energy flux into a domain with an inner boundary of any radius $r=r_1$ equals the flux out of that domain bounded by any outer, larger radius $r=r_2$. It agrees with the fact that $ {\partial } E/ {\partial } t=0$ and shows that indeed $\boldsymbol {\nabla }\boldsymbol {\cdot } {\boldsymbol {F}}=0$.

Finally, let us note that the time-integrated radial energy flux as a function of direction (polar angle $\theta$) is indeterminate because the integral of the isotropic part, the first term in (5.8), diverges. But the anisotropic part $\int _0^{\tau } \mathcal {F}(r,\theta ,t)\,\textrm {d} t$ of the radial flux can be calculated. It is shown in figure 8 for time $\tau =50$. We find that as $\tau$ increases, the undulations in this curve, representing contributions of individual Rossby waves, vanish. The dominant peak in the total flux toward the western side of the forcing is obvious in figure 8.

Figure 8.

Azimuthal distribution of the time-integral of the anisotropic part of the radially outward-directed energy flux $F \equiv \int _0^{\tau } \mathcal {F}(r,\theta ,t)\,\textrm {d} t$ through the unit circle $r=1$. Here, $\mathcal {F}$ is defined in (5.8). The centre of this graph $\theta ={\rm \pi}$ corresponds to west. The example is for $\tau =50$.