Our model domain is a rectangle of length Lx and width Ly, extending from x=0 to x=Lx in the zonal direction and from y=-Ly/2 to y=Ly/2 in the meridional direction. In the entire paper, the length of the domain is set to 1Lx=2πacos⁡ϕ0, where a=6371.2 km denotes the radius of the Earth and ϕ0=45° N is a reference latitude. For terminological convenience we will refer to the zonal direction as “longitude” and the meridional direction as “latitude”, although we stick to Cartesian geometry throughout the paper. The beta-plane approximation implies that the Coriolis parameter is given by 2f(y)=f0+βy, with f0=2Ωsin⁡ϕ0 and β=2Ωa-1cos⁡ϕ0.

We assume a basic state that is zonally symmetric and purely zonal, but its zonal wind u0(y) may depend on latitude. Linearizing the inviscid barotropic vorticity equation about this basic state and assuming some external stationary forcing F′, one obtains 3∂∂t+u0∂∂xq′+q0yv′=F′, where q′ is perturbation absolute vorticity, v′ is the perturbation meridional wind, and 4q0y=β-d2u0dy2 is the meridional gradient of the basic state absolute vorticity.

The forcing F′ is modelled through a dimensionless pseudo-orography h′(x,y) as 5F′(x,y)=-f0u0∂h′∂x. The term “pseudo-orography” indicates that the so-obtained vorticity source simulates the effect of orography within the limited framework of the barotropic model . In the entire paper we only consider pseudo-orography which is sinusoidal in longitude, i.e., 6h′(x,y)=ℜh^(y)eikx, where ℜ… denotes the real part and h^(y) characterizes the meridional profile of the orography.

In most parts of this paper the meridional profile h^(y) is assumed to be “meridionally thin”. For the analytical treatment it is represented by a delta function like 7h^(y)=Dδ(y), with D=500 km, and this will be referred to as delta-forcing in the following. Note that the delta-function has units of m-1 such that h^ and h′ are, indeed, dimensionless. For our numerical solutions, Eq. () is replaced by 8h^(y)=DD̃cos⁡2π2D̃yfor|y|≤D̃, and zero otherwise. Unless stated otherwise we use D̃=500 km, and in all our model configurations we satisfy D̃≪Ly. This guarantees that h^(y) from Eq. () is “merdionally thin” and can be taken as approximation to the delta function. At the same time, D̃ is always chosen wide enough such that the non-zero part of h^(y) in Eq. () can be represented by a fair number of grid points and, hence, properly resolved in our numerical treatment. Note that for both Eqs. () and () one obtains 9∫-Ly/2Ly/2h^(y)dy=D, which means that the amplitude of the orography is equivalent in an integrated sense.

Writing q′ and v′ in terms of the perturbation streamfunction ψ′, 10q′=∇2ψ′,v′=∂ψ′∂x, and restricting attention to stationary solutions, one obtains 11u0∂∂x∇2ψ′+q0y∂ψ′∂x=-f0u0∂h′∂x. We look for solutions of the following form 12ψ′(x,y)=ℜψ^(y)eikx, divide by u0, and obtain 13d2ψ^dy2+q0yu0-k2ψ^=-f0h^. In the remainder of this paper we only consider basic states satisfying u0>0 throughout the interior of the domain such that Eq. () is free of singularities.

For later reference we define the square of the stationary wavenumber 14Ks2=q0yu0 and the dimensionless stationary wavenumber 15K^s=Lx2πKs2. For a constant basic state wind, both Ks2 and K^s are constant, but for more general profiles of u0(y) they are functions of latitude. A typical mid-latitude jet satisfies Ks≲10 within the jet region .

Introducing the dimensionless zonal wavenumber 16s=Lx2πk, Eq. () can be rewritten as 17d2ψ^dy2+2πLx2K^s2-s2ψ^=-f0h^. Considering the value Lx as given and fixed, the above equation indicates that the local character of the solution ψ^ outside the forcing region only depends on the function K^s(y) and the value of s. In particular, the solution has an oscillatory character for latitudes where K^s2>s2, while is has an exponential character for latitudes where K^s2<s2. It follows that the soliution ψ^ does not necessarily satisfy ψ^=0 where K^s2=s2.

The meridional component of the linear wave activity flux is given by 18F(y)=-u′v′‾, where the overbar denotes the zonal average. For solutions with a fixed zonal wavenumber, this can be reformulated in terms of the perturbation streamfunction ψ′=ℜψ^(y)exp⁡(ikx) as 19F(y)=12ℜikψ^dψ^∗dy, where the asterisk denotes the complex conjugate. Assuming furthermore that the zonal wavenumber k is real and ψ^(y)∝exp⁡(ily), the last expression turns into 20F(y)=12k(ℜl)|ψ^|2; in this case the meridional flux of wave activity vanishes if l=0 or if l is purely imaginary.

Periodicity of the domain sets a constraint on k, namely that the zonal wavenumber must be quantized according to kLx=!2πs or 21k=!2πLxs. with s=1, 2, 3, …. The integer s represents the number of entire wavelengths that fit into the domain in the zonal direction.

At both the southern and the northern boundary of the domain we posit that a certain fraction R of wave amplitude is reflected, while the remaining part is transmitted. Following their Eq. 18, this can be achieved by specifying 221-R1+Rdψ^dy=±iq0yu0-k2ψ^aty=±Ly/2. Note that this boundary condition is singular in the limit R→1, and one obtains the familiar Dirichlet condition ψ^=0 for fully reflecting conditions R=1 (except when the square root on the right hand side happens to be zero). In the remainder of this paper, the special model configuration with R=1 will be referred to as a “reflecting periodic channel”. The condition ψ^=0 in this case represents another quantization constraint, namely that an integer number n of half wavelengths must fit into the meridional extent of the channel, i.e., 23l=!nπLy with n=1, 2, 3, ….

Equations () and () represent a 1D boundary value problem, which can be solved numerically in a straightforward manner. The differential operator on the left hand side of Eq. () is discretized using standard finite differences, reducing the solution for the interior grid points to the inversion of a square matrix. The boundary conditions are implemented by either (in case of R=1) setting the boundary grid points of ψ^ to zero, or (in case of R<1) by modifying the equations for the interior grid points such as to account for the discretized version of Eq. (). The resulting square matrix is inverted using a linear algebra routine from scipy.

In case of the forced harmonic oscillator from theoretical physics, there is a straightforward recipe to diagnose resonant behavior: try many different values for the forcing frequency (using identical forcing amplitude) and determine whether and to what extent the stationary reponse shows a pronounced peak in amplitude in the neighborhood of a specific forcing frequency.

Our strategy to diagnose Rossby wave resonance closely follows this idea: we compute the stationary solution ψ′ for an entire range of zonal wavenumbers s (with the same forcing amplitude, i.e., the same value of D for each value of s) and evaluate how different aspects of the solution change as a function of s. One particular “aspect” of the solution is, obviously, its amplitude: to the extent that the amplitude shows a pronounced peak at one or several specific values of s, we associate the basic state with resonant behavior at these values of s. In fact, for the current purpose we can consider s to be a positive real number (rather than an positive integer), because the solution for the meridional structure problem is effectively ignorant of the boundary conditions in the zonal direction. In addition, we analyze the solution's phase as a function of s, because the phase behavior serves as another hallmark of resonance .

In the following two section we are going to consider various model configurations that differ in the basic state zonal wind u0(y) and in the choice of the boundary conditions. For illustration the reader is refered to Fig. . The configurations depicted in Fig. a, b, and c allow analytical solutions (Sect. ), while the configuration in Fig. d requires one to resort to the numerical solution procedure (Sect. ).

The search for free modes (or normal modes) of a linear system is motivated by the recognition that resonance occurs if the forcing projects onto a stationary free mode. Free modes are solutions of Eq. () with F′ set to zero. We restrict attention to perfectly reflecting channel walls in this subsection. The model configuration corresponds to the situation depicted in Fig. a, except that there is no forcing. With these assumptions, there is a discrete but infinite set of solutions, namely 24ψn,s′(x,y,t)=ℜψ0eik(x-ct)sin⁡ly+Ly2, where the wavenumbers k and l are limited to discrete values given by Eqs. () and (), respectively, and where the phase speed c satisfies the well-known dispersion relation 25c=U-βk2+l2. Apparently, the free modes are quantized not only in the zonal direction (due to the requirement of periodicity, non-dimensional wavenumber s), but also in the meridional direction (due to the finite width of the reflecting channel, non-dimensional wavenumber n). For illustration, we show two examples in Fig. , namely ψ(4,3)′ (associated with c=2.0 m s⁻¹) and ψ(2,4)′ (associated with c=-3.7 m s⁻¹). The corresponding modes on the sphere are the so-called Rossby-Haurwitz waves .

The discrete set of normal modes can be represented as points on the (k,l)-plane. This is done in Fig.  (blue points) for three different values of the channel width Ly. The horizontal rows of points in this diagram represent modes with the same value of n but varying value of s, and the value of n increases from the bottom row to the top row. Since we assume that Lx is given and fixed, the distance between the horizontal rows of points depends on the value of Ly according to Δl=π/Ly.

Most of the modes displayed in Fig.  are associated with a nonzero phase speed c according to (). The solid red line in this plot depicts the location in wavenumer-space where c=0, which is equivalent to 26k2+l2=βU≡Ks2. All modes that lie to the bottom-left of the red circle have c<0, while all modes that lie to the top-rigtht of the red circle have c>0. Particularly interesting are those few modes that happen to lie on (or are very close to) the red circle, because they are (almost) stationary. These modes are associated with resonance if an appropriate stationary forcing is switched on. With “appropriate” we mean that the forcing has a non-vanishing projection onto the respective mode.

As was mentioned before, a westerly jet can act as a waveguide, although its waveguidability is likely to be less than 1. Leakage across the jet flanks can, to some approximation, be considered as similar to damping, and even in a leaky channel one may obtain a peak in amplitude as one moves across the resonant wavenumber . Hence, a reflecting channel can be informative.

The important point here is that out of the three options for Ly displayed in Fig. , only the choice in Fig. a can be taken as representative for a midlatitude jet. By contrast, the channel widths in Fig. b and c are much larger than the width of a typical jet. After all, the defining characteristic of a midlatitude jet is that its zonal scale is much larger than its meridional scale. As a consequence of this anisotropy, the red circle in Fig. a has only one intersection with the light blue line, suggesting the existence of just one resonant peak. Note that for an even smaller value of Ly (not shown) there may in fact be no intersection at all between the red circle and any of the blue lines. It transpires that for a jet-typical scenario, only the first meridional mode (n=1) is likely to contribute to resonant behavior. In other words, higher meridional modes (n>1) can contribute to resonance only in channels with unrealistically large width (Fig. b and c). This suggests that our strategy with varying s and checking the result should yield in no more than one resonant peak for any realistic jet width – and (as we will see) this is what we obtain in most cases. The location of the peak should correspond to the intersection of the red circle with the light-blue line in Fig. a, which in our example is at s=3.25.

More formally, the condition for resonance is Eq. (). Accounting for the quantizition () in the meridional direction, but considering s as continuous, one obtains the following expression for the resonant wavenumber 27sres=Lx2πKs2-l2≡K^s2-nLx2Ly2, with n=1, 2, 3, …. Of course, the resulting values sres would be integers only by chance. Yet, to the extent that sres is close to an integer for one or several values of n, the free mode is close to resonance and one may expect that the corresponding forced solution has a large amplitude. In addition, the requirement that sres must not be imaginary restricts the set of allowed values of n in the above relation. Given that a jet is characterized by Lx≫Ly and that typically K^s<10, it transpires that the meridional mode n=1 is likely to be the only one that is associated with a physical (i.e., non-imaginary) value for sres.

We now keep the same configuration as in the previous subsection except that we switch on forcing of the following form 28h^(y)=h0cos⁡ly,withl=π/Ly. This type of forcing was used a long time ago by and will, hence, be referred to as Charney-Eliassen forcing. The resulting model configuration is illustrated in Fig. b. Note that the Charney-Eliassen forcing is less general than the delta-forcing in the sense that it contains only one specific meridional wavenumber. By contrast, the Fourier-decomposition of the delta-funcion contains all possible wavenumbers, and the solution is freer to “choose” its meridional wavenumber.

We are looking for stationary solutions with sinusoidal shape in the zonal direction. The relevant equation to be solved is Eq. (). Using the Ansatz ψ^=ψ0cos⁡ly (which satisfies the fully reflecting meridional boundary condition), one obtains 29ψ^(y)=f0h0K2-Ks2cos⁡ly and, hence, 30ψ′(x,y)=f0K2-Ks2h′(x,y), where K=k2+l2 denotes the total wavenumber. Apparently, the amplitude of the response ψ′ is proportional to the strength of the forcing h′, which is a generic property of any linear forced system. What's more interesting is the denominator on the right hand side. The latter turns zero and, hence, the response turns infinite when the total wavenumber equals the stationary wavenumber, K2=Ks2, or equivalently 31k2+l2=βU. This singularity is arguably a hallmark of linear resonance. Obviously, Eq. () is equivalent to the condition () for the corresponding normal mode to be stationary. In addition, the sign of ψ′ switches discontinuously for increasing k or s as one moves across the singularity, and this corresponds to a phase change by π. Such a phase change between the forcing and the response is another hallmark of linear resonance.

According to Eq. (), the meridional wavenumber l in the Charney-Eliassen configuration is determined by the meridional channel width. Correspondingly, the condition for resonance becomes 32s=!sres=Lx2πβU-πLy2. It follows that there is either one resonant wavenumber or none, depending on whether the expression under the square root is positive or negative. Regarding the existence of free modes as illustrated in Fig. , one obtains only one row of blue points corresponding to n=1, allowing either one or no intersection between the light blue line and the red circle. Basically, the specific meridional shape of the Charney-Eliassen forcing () excludes the higher meridional modes from the solution.

If one chose to include damping by adding -αq′ (with α>0) to the right hand side of Eq. (), one would obtain an additional, purely imaginary term in the denominator on the right hand side of Eqs. () and (). It follows that damping prevents the singularity; yet, for small enough values of α the functional dependence of |ψ^| on k or s still shows a pronounced peak close to the resonant wavenumber.

At first sight it seems that the Charney-Eliassen model configuration is not well suited to investigate Rossby wave resonance along a jet, because it makes two rather strong assumptions. First, the forcing has a very specific structure in the meridional direction, necessitating the same specific structure for the solution ψ′; this may be considered as dangerous, because more general forcing may project onto higher meridional modes, and it is not entirely clear at this point how this would affect the solution. Second, the Charney-Eliassen solution assumes perfectly reflecting channel walls; as was shown by , this assumption must be considered as unrealistic, because Rossby wave resonance on a jet is more akin to resonance in a channel with some leakage of wave activity across the channel walls. These two issues motivate the following modified model configuration as a better alternative: instead of Charney-Eliassen forcing we now use our delta-forcing as defined in Eq. (), and we furthermore allow some leakage by specifying R<1 at the channel walls. The model configuration for this set of experiments is illustrated in Fig. c.

Analytical progress can still be made by sticking to a constant wind U. In this case, the solution for either y>0 or y<0 is a superposition of plane waves like ψ′∝exp⁡i(kx+ly) with k and l satisfying Eq. (). The coefficients must be determined through a matching condition at y=0, which effectively accounts for the forcing. Using very similar methods as in , we obtain 33ψ^(y)=Aeily+Be-ily,y≥0Beily+Ae-ily,y<0 with 34A=if0D2l1+ReilLy,35B=-iRf0DeilLy2l1+ReilLy=-ReilLyA, and with D as defined in Sect. . For a fixed value of k, the meridional wavenumber is given by virtue of Eq. () as 36l=βU-k2. Hence, for any given U, the value of l depends on k or s, respectively, and the solution ψ^(y) depends on k or s in a nonlinear fashion through Eqs. () and (). Considering the zonal wavenumber as continuous, the relation () does not represent a quantization constraint for l, in contrast to Eq. ().

The above solution suggests resonant behavior when the denominator in the expressions for A and B vanishes, i.e., when 37l1+ReilLy=0. For fully reflecting boundaries (R=1) this condition can be satisfied through the second factor on the left hand side. It requires l to be real and to satisfy the following quanitzation rule 38l=!nπLy,n=1,3,5,… By means of Eq. (), the above translates to a condition for s, namely 39s=!sres≡Lx2πβU-nπLy2 with n=1, 3, 5, …, where the choice of admissible values for n is limited through the condition that s must be real. By contrast, for partial reflection (R<1), there is no true singularity, although the solution still may have a pronounced peak at the values of l given in Eq. () to the extent that R is close to 1.

Comparison of Eq. () with the corresponding condition () for the Charney-Eliassen configuration indicates that the latter is a special case of the former: Charney-Eliassen only accounts for n=1, while the current relation possibly allows higher meridional modes with n>1.

Interestingly, condition () resembles, yet is different from, the condition () for the existence of normal modes. More specifically, the resonant modes of our current problem correspond to only the odd meridional normal modes. The reason lies in the fact that all even modes have a node at mid-channel latitude, and this is exactly where our delta-forcing is located. It follows that the special form of our forcing in Eq. () allows a non-zero projection only onto the odd modes and can, therefore, trigger resonance only for this reduced set of modes.

In addition, condition () is satisfied when l=0, and formally this corresponds to n=0 in Eqs. () and (). In this case, the meridional flux of wave activity vanishes owing to Eq. (), which means that wave activity is ducted in the zonal direction. Again, this should result in resonant behaviour thanks to the zonal periodicity as soon as s is an integer. We will refer to this solution as the n=0 meridional mode.

The two options allowing resonance are distinctly different, because the first option includes meridional wave propagation while the second does not. However, the only aspect that is relevant for resonance is the fact that wave activity is channeled in the zonal direction without leakage in the meridional direction, and this is guaranteed for both options. In the first option it is achieved through the existence of perfectly reflecting meridional boundaries, while in the second option it is achieved through the flux of wave activity being purely zonal.

We now follow our general strategy and test for resonant behavior by varying s and analysing both the amplitude and the phase of the stationary solution. The result is shown in Fig.  for various values of R and with U and Ly fixed at U=10 m s⁻¹ and Ly=3000 km, respectively. The amplitude of the response is measured as max⁡y|ψ^(y)| and its phase as the phase of ψ^ at the jet latitude. For the current choice of parameters, condition () predicts two resonant peaks, one at s=3.25 corresponding to the first meridional mode n=1, and one at s=5.73 corresponding to n=0. We first consider the behavior at s=3.25. Apparently, the amplitude for R=1 in Fig. a indicates a singularity at this value of s, while for smaller values of R the peak gets less pronounced and vanishes completely for values R≲0.25. The singularity in amplitude at s=3.25 in Fig. a is mirrored by the behavior of the phase in Fig. b; in particular, the phase is discontinuous at s=3.25 for fully reflecting conditions (R=1), giving way to a more gradual transition for R<1. This general behavior in terms of amplitude and phase is very similar to the damped linear oscillator from classical mechanics; furthermore, it is consistent with , who showed that partial transmission at the channel boundaries has a similar effect on resonance as damping.

The other option for resonance implies n=0, which for the current choice of parameters occurs at s=5.73 according to Eq. (). Indeed, there is a pronounced (but very narrow) peak in Fig. a at this location, at least for R<1. Interestingly, this peak is absent for R=1. Further investigation (see Appendix) reveals that the limit R→1 for the n=0 resonance is singular: although each of the coefficients A and B in Eqs. () and () blow up individually for l→0, the sum of both terms on the right hand side of Eq. () remains finite. The singularity in amplitude is reflected by a special behavior of the phase, which shows a jump by π/2 across s=5.73 for R=0 and a more complicated behavior for 0<R<1 (Fig. b). Interestingly, the value π/2 differs from the value π that one obtains in case of the harmonic oscillator from classical mechanics. We speculate that this is related to the fact that there is an asymmetry as one moves across s=5.73: the system supports free Rossby waves for s<5.73, while it does not support free Rossby waves for s>5.73.

We further illustrate the analytical solution for a number of parameter choices in Fig. . This figure shows the patterns of the perturbation streamfunction on the longitude-latitude plane for three different values of R and three different values of s. First we note that fully reflecting channel boundaries (top row) always imply ψ′=0 at the channel walls – by design. There is no phase tilt with latitude, because the northward and the southward traveling wave have equal amplitude. In all other cases with R<1, the solution features non-zero values at the channel walls. There is a meridional phase tilt in Fig. d and g close to the channel walls consistent with outward wave propagation. However, this phase tilt vanishes for s≥K^s (second and third column), because in this case the meridional wavenumber l is either zero (second column) or imaginary (third column) owing to Eq. (); this situation is equivalent to no meridional wave propagation according to Eq. ().

The three panels in the left column of Fig.  are particularly relevant for our further analysis. They represent situations which do allow meridional wave propagation. Proceeding from the top to the bottom of this column, one can identify a noteworthy transition from modal behavior for fully reflecting conditions (Fig. a) to plane-wave behavior for fully transparent condition (Fig. g). Unsurprisingly, “modal behavior” is qualitatively reminiscent to the normal modes of Fig. . The intermediate situation for R=0.5 (Fig. d) looks like a superposition of the two extreme cases and, thereby, contains aspects from both. We anticipate that the intermediate case will help to understand more realistic jet profiles u0(y), because (as we will see) these are associated with partial reflection and partial transmission of wave activity at the periphery of the jet flanks.

We emphasize, again, the different nature of the two resonant peaks in Fig. a. While the n=1 resonance (at s=3.25) produces a sharp peak for R→1, the n=0 resonance (at s=5.73) produces a peak for any value of R except R=1. As mentioned earlier, for the n=1 resonance the waves travel both northward and southward and keep superimposing if the meridional wavelength happens to be just right. By contrast, for the n=0 resonance the meridional flux of wave activity is zero such that wave activity that is generated at y=0 cannot escape in the meridional direction and, hence, keeps accumulating within the domain. In both cases there is a physical mechanism that prevents leakage in the meridional direction and, hence, allows resonance.

We also show results for other choices of the channel width (Fig. ). First consider Ly=1500 km in Fig. a. Apparently, for such a narrow channel we only obtain the peak corresponding to n=0. By contrast, increasing the value of Ly to 10 000 km (Fig. b) makes the resonant peaks for n=0 (at s=5.73) and n=1 (at s=5.55) almost coalesce, and one obtains a third peak at s=3.85 corresponding to n=3.

Let us briefly restrict attention to the perfectly reflecting channel (black line in Fig. b) and further illuminate these results by connection with the idea of resonant normal modes as illustrated in Fig . In the latter figure, the spacing between the horizontal light-blue lines increases as the channel width decreases. It follows that the possibility for resonance completely vanishes when the channel becomes too narrow. On the other hand, for Ly = 10 000 km (Fig. b), there are multiple intersections between the red circle and the horizontal light-blue lines; the intersections with the first and the third light-blue line from below correspond to the peaks n=1 and n=3 in Fig. b. If one chooses an even larger (and clearly unrealistic) channel width (Fig. c), one obtains a very large number of intersections between the red circle and the different light-blue lines. Translated to our strategy of diagnosing the amplitude as a function of continuous s, this would produce considerably more (and more densely spaced) peaks compared to those in Fig. b. In the limit Ly→∞, literally every real value for s would be associated with resonance.

We start with validating our numerics by considering a constant basic state u0=U=const and comparing the numerical solution with the corresponding analytical solution. Using Ly=3000 km, we obtain the result shown in Fig. . Comparison with Fig.  indicates that the overall behavior is very similar. In particular, the location of the resonant peaks in Fig. a is exactly where expected from Eq. () with n=0, 1, 3, 5, …, and the dependence on R in both panels of Fig.  is qualitatively similar as in the analytical solution from Fig. . Admittedly, the numerical solution does not quite reproduce the exact behavior in the neighborhood of the n=0 resonance. A closer examination indicates that this is presumably due to the finite meridional width of the forcing in the numerical model configuration. Overall, however, we consider the agreement between the analytical and the numerical solution as very satisfying, thus providing credibility to our numerics.

We now turn to the core of our analysis and consider more realistic jet-like wind profiles u0(y). The model configuration for this set of experiments in illustrated in Fig. d. The latitudinal variation of u0(y) precludes a general analytical solution, but the numerical solution remains straightforward.

In contrast to earlier, we now restrict our attention to fully transparent boundary conditions at the meridional boundaries. Basically, we aim to learn whether and to what extent the flanks of the jet themselves have partly reflecting properties, and this would be confounded if we included reflection at the meridional boundaries. We posit that any amount of wave activity that manages to escape the jet region can freely propagate away towards infinity in the meridional direction. Hence, we set R=0 at the meridional boundaries as a natural choice for this set of experiments. We also make sure that the entire jet is contained in our computational domain, which means that u0 transitions to a practically constant wind profile close the meridional boundaries. Incidentally, for any such wind profile our results should not depend on the meridional width of the domain owing to the fully transparent boundaries. We checked this prediction and found that, indeed, the numerical solutions are practically independent of the choice of Ly.

In all considered cases, our background wind is specified to be a Gaussian westerly jet superimposed on a constant U>0, 40u0(y)=U+(umax-U)e-y2/(2σJ2), with U>0 and umax≥U, and where σJ represents the width of the jet. This choice implies u0>0 everywhere and, thus eliminates any possible singularity in Eq. (). In addition, it implies that barotropically unstable modes must have a positive phase velocity according to Howard's semicircle theorem e.g.; our focus on forced stationary solutions thus excludes barotropically unstable modes.

Of course, instability may be another important mechanisms for wave growth – be it barotropic instability in the barotropic model, or baroclinic instability in a more realistic framework. However, in connection with Rossby wave resonance in observed episodes the focus is often on stationary modes, since these are more likely to be associated with extreme weather than traveling modes . In the past, this focus was achieved through time averaging, like, e.g., by analysing monthly means or by preprocessing the data with a 15 d running mean . The focus on stationary modes is straightforward in our linear framework, because modes of different phase velocity are independent.

In our attempt to understand the transition between a constant basic state and a jet-like basic state, we change the amplitude of the jet but keep its width constant at σJ=500 km. The three wind profiles used are shown in the top row of Fig. . The first one in Fig. a is a constant wind u0=U=10 m s⁻¹ (as before, i.e., no jet at all), the second one in Fig. b is a weak jet with umax = 16 m s⁻¹, and the third one in Fig. c is a strong jet with umax = 30 m s⁻¹.

The resulting resonant behavior is presented in the third row of Fig. . In all three cases there is only one single peak. The interpretation of the peak in Fig. g is straightforward thanks to our previous analysis of the analytical solution: the sharp peak represents the n=0 resonance (see the blue line in Fig. a), and there cannot be any further peak representing n>0 because of our fully transparent meridional boundaries. Interestingly, both the location and the character of the peak changes as one proceeds from the constant U to the strong jet (Fig. g, h, and i). Based on the experience from the previous section, the somewhat more gradual shape of the peak in Fig i is reminiscent of a n=1 resonance in a partly reflecting channel. This interpretation is supported by the phase behavior in the second-to-last row (Fig l versus Fig j). At the same time, the weak jet in the middle column of Fig.  represents a situation which is intermediate between the constant wind and the strong jet.

We designed a metric Q which is meant to measure the “quality” or “strength” of the resonance. This is achieved by quantifying the sharpness of the peak in the functional dependence of f(s)=(max⁡y|ψ^|)(s) as shown in the third row of Fig. . We first determine sres as the wavenumber at which the function f(s) maximizes, and then we define 41Q=2f(sres)f(sres-1)+f(sres+1)-1. The diagnostic is designed such that Q=0 if f(s) is a constant function, and Q increases to the extent that the maximum represents an increasingly narrow peak. The value Q=1 is reached when (for a symmetric peak) the maximum value is twice as large as the ambient values at a distance Δs=±1. This value (Q=1) can be taken as a meaningful threshold for the occurrence of resonance in meteorological applications. The Q-values for the three basic states in Fig.  are provided in the respective panels in the third row. Apparently, the n=0 peak in the left column is very sharp resulting in a very high value Q≫1. By contrast, the weak jet is associated with a value Q<1, while the strong jet has Q>1. The latter behavior is consistent with the conventional wisdom that stronger jets are better waveguides, implying a stronger tendency for resonant behavior.

Can we “understand” the transition between the solutions shown in the three columns in Fig. ? The n=0 resonance visible in the left column gets weaker to the extent that the wind profile has an increasing amount of latitudinal variation, and this happens presumably for two reasons. First, the value of sres in Eq. () for n=0 turns less well defined to the extent that the basic state wind is not a constant any longer. At the same time, the latitudinal variation of u0 prevents a unique value of l in Eq. (). As a consequence, the condition l=0 can be satisfied only at one specific latitude rather than within a whole range of latitudes. Second, we posit that the flanks of a jet are associated with at least partial reflection R>0, with increasing values of R for increasing jet strengths . Figure a suggests that the width of the n=0 resonant peak decreases as the value of R increases, and this means that the n=0 resonant peak gets less dominant. At the same time, as one starts to build a jet, this jet is associated with an increasing amount of reflection, and one starts to obtain an n=1 resonant peak; the strength of this peak should increase for stronger values of R (Fig. a) and, hence, for stronger jets. This interpretation is supported by the fact that the peak in Fig. i is rather wide, while the peak in Fig. g is very sharp – consistent with the different shapes of the peaks for n>0 versus n=0 in our analytical solutions from the previous section. In summary, we suggest that there is a gradual “blend-out” of the n=0 peak and a gradual “blend-in” of an n=1 peak as one proceeds from the constant wind to the strong jet. Apparently, the solution for the weak jet in the middle column of Fig.  lies somewhere in between these two extremes.

The interpretation offered above is consistent with the patterns of the perturbation streamfunction for the three solutions (bottom row in Fig. ). The constant basic state (Fig. m) shows – by design – the behavior from the corresponding analytical solution (Fig. h). The other two scenarios (Fig. n and o) show an increasing amount of confinement of wave activity to the jet-region, with outgoing plane waves beyond the jet region. In particular, the pattern in Fig. o is reminiscent of the pattern of the analytical solution for a partly reflecting (or partly leaking) channel from Fig. d. Note also, that the wave pattern in Fig. o resembles the jet pattern in Fig. c regarding its meridional structure.

We now aim to corroborate the interpretation of the resonant behavior for the strong jet case (right column of Fig. ) in terms of approximate partial reflection at an internal interface at the periphery of the jet flanks. As mentioned earlier in connection with Eq. (), the two key variables in this equation are K^s and s, determining whether the local character of the solution is wavelike or exponential. Let us apply this diagnostic concept with the help of Fig. e and f. For those values of s2 that lie between the relative maximum of K^s2 at the jet core and the relative minimum of K^s2 at the jet's flank, the character of the solution switches from wavelike in the jet core to exponential at the jet periphery. The stronger the jet, the larger is the corresponding range of wavenumbers. We hypothesize that the “exponential regions” at the jet's periphery act as partial reflectors and, hence, generate a certain amount of waveguidability.

Let us shed more light on this hypothesis. We assume that for both subdomains y>0 and y<0 the total perturbation streamfunction consists of two parts: the transmitted part ψtrans′ which is able to leave the domain, and the remainder ψrefl′ which participates in the reflection, i.e., 42ψ′=ψrefl′+ψtrans′. Close to northern meridional boundary, the transmitted part has the form of an outgoing plane wave, i.e., 43ψtrans′=ψ^transei(kx+ly). This part of the solution is obtained by, first, computing l from Eq. () with u0(±Ly/2) substituted for U, and then inferring ψ^trans from the known full solution at the domain boundaries. The reflected part ψrefl′ is then obtained from Eq. (). This procedure is carried through separately for the two subdomains y>0 and y<0.

The pattern of the resulting ψrefl′ is shown in Fig.  for the two jet-like wind profiles from the middle and right column in Fig. . Apparently, there is very little phase tilt with latitude in ψrefl′. The behavior is consistent with the notion that this part of the solution is associated with reflection at a latitude somewhere between the middle of the domain and the domain boundaries, resulting in a modal pattern of streamfunction (cf. Fig. ). In fact, one may associate the reflected part ψrefl′ with an effective channel width by estimating the latitudes at which the amplitude goes to zero. In Fig. b, this happens at y≈±1200 km; thus the effective channel width associated with this jet is Ly≈2400 km. Note that this value is considerably larger than the meridional extent of the “jet cavity” where K^s2≥sres2, and that this is expected according to what we mentioned in the text behind Eq. (). Furthermore, the stronger jet (Fig. b) is associated with a considerably stronger ψrefl′ than the weaker jet (Fig. a), and this is consistent with the accepted wisdom that stronger jets are a better waveguides.

To the extent that our strong jet produces partial reflection and a mode-like behavior in the core of the jet, we should be able to relate this interpretation to the analytical solution in a reflecting channel with constant U. More specifically, we aim to predict the resonant wavenumber from Eq. (). Using the above estimate of the effective channel width Ly≈2400 km and the value of K^s≈7 close to the jet core (Fig. f), we obtain a single resonant wavenumber for n=1 at sres≈3.8, which happens to be identical to the diagnosed value of sres=3.8 in Fig. i. To be sure, the estimated value of sres sensitively depends on the chosen values for Ly and K^s, none of which are well-defined in the jet-scenario. Yet, we consider this result as a “sanity-check”, adding confidence to our interpretation in terms of partial reflection at a latitude close to the periphery of the jet's flanks.

It is also illuminating to shift the orography in the meridional direction away from the jet. More specifically, we extend the southern part of the domain to y=-5000 km and shift the pseudo-orography to y=3000 km. The forcing thus lies outside of the jet, in a region with constant wind u0=10 m s⁻¹. For any s<5.7 we expect that locally there must be two plane waves emanating from the new forcing location similar as in Fig. g. However, the wave that travels northward is going to encounter the jet. Based on the earlier results from this subsection, one may expect multiple reflection between internal interfaces located at the periphery of the jet flanks such that only part of the wave activity is able to escape the jet region and leave the domain through the northern boundary. If the zonal wavenumber of the forcing happens to be s=3.8, these multiply reflected waves should interfere constructively resulting in increased wave amplitudes at the jet latitude. Indeed, this is exactly what our numerical solution shows (Fig. ). Apparently, the presence of the jet is able to generate a modal structure within the jet region; this effectively channels wave activity in the zonal direction and, thus, leads to increased wave amplitudes owing to repeated superposition thanks to the periodic boundaries. Note that the magnitude of this maximum response in the jet core turns out to be considerably smaller (viz., only one third) compared to the value obtained in Fig. i; however, this is qualitatively to be expected, because the shifted forcing has a smaller projection on the resonant mode (cf. Fig. o).

Finally, we consider the resonant behavior for jet-like wind profiles as the width σJ of the jet varies while its amplitude umax is kept constant (Fig. ). In this set of experiments, both the domain width Ly and the value of D̃ in Eq. () are varied in the same proportion as σJ; this device guarantees that the meridional extent of the orography is always considerably smaller than the jet width and, at the same time, the orography is numerically well resolved in each experiment.

There is a wide range of behavior of the refractive index across the three experiments depicted in Fig. . The function K^s(y) shows a relative maximum at the jet latitude for the narrow and the intermediate jet, but a relative minimum for the wide jet (note that the our wide jet is unrealistic in the sense that it would not fit into the midlatitudes on planet Earth). The relative minimum in this case can be explained by noting that in the wide-jet limit the second derivative of the wind profile can be neglected in Eq. (); the stationary wavenumber squared becomes 44Ks2(y)≈βu0(y), from which one obtains a local minimum of K^s(y) at the jet core. By contrast, in the narrow jet limit the stationary wavenumber squared scales like 45Ks2≈-1u0d2u0dy2∼1Δjet2. In this case one expects a sharp relative maximum of K^s(y) at the jet latitude that scales like Δjet-1.

In case of the wide jet, K^s=3.5 in the jet core (Fig. f), and this value is identical to the location of the resonance (sres=3.5, Fig. i). In other words, sres=K^s(0), and this implies n=0 according to Eq. (). Indeed, this result is broadly consistent with the qualitative behavior of the phase (fourth row), which suggests a transition from an n=1 resonance for the intermediate jet to an n=0 resonance for the wide jet. Moreover, as we increase the width of the jet even further (not shown), the quality-measure Q of the resonant peak increases substantially, consistent with the very sharp peak of our analytical solution in the constant wind case with R=0 (blue line in Fig. ). As we will argue below, the scenario of our wide jet is unlikely to be relevant in practice, but the consistent interpretation is nevertheless satisfying.

By contrast, the solution for the intermediate jet in the middle column of Fig.  has the flavor of a first (n=1) meridional mode. As discussed in the previous section, this mode is established through reflection of wave activity at the periphery of the jet flanks. The fact that the resonant peak is located at a very similar wavenumber for the intermediate and for the wide jet (Fig. h and i) must be considered as fortuitous: apparently, the change of K^s(y) and the change of the “effective channel width” nearly compensate each other.

Let us finally turn to the narrow jet in the left column of Fig. . Figure g shows a single resonant peak at roughly the same wavenumber as for the intermediate jet (Fig. h), but the sharpness (i.e., the Q-value) of the peak is considerably lower. The weakness of the resonance appears plausible in view of Fig. : given that the width of an “equivalent reflecting channel” would be only about 1000 km, this should actually prevent the existence of a stationary normal mode and, hence, the occurrence of a well-defined resonant peak. Of course, this argument has to be taken with a grain of salt, since the narrow jet cannot possibly be modelled through a constant wind in a quantitative manner.