On the Effect of Surface Friction and Upward Radiation of Energy on Equatorial Waves

a. Strict quasi-equilibrium troposphere

KE18 and E20 derive and analyze an equation set for a linear system that describes the dynamics and thermodynamics of an atmosphere that maintains a vertically constant saturation moist entropy $s^{\ast }$ in the free troposphere. However, in both of those studies, the authors assume a rigid lid and frictionless surface. Hence, in their model, only the baroclinic mode can be excited, and upward radiation into the stratosphere is absent.

Equations (14)–(17) and (19)–(21) formulate the tropospheric system, where u₀, υ₀, u₁, υ₁, ϕ₀, $s^{\ast }$, and s_m are the unknown variables. Note that the linear system is not complete: additional specification of the vertical velocity at the tropopause is required to complete the system. As mentioned previously, studies that assume a rigid lid set the tropopause velocity to be zero. Other studies parameterize the tropopause dynamics using a wave-radiation upper boundary condition (Moskowitz and Bretherton 2000; Kuang 2008a). In this study, we take a different approach and couple the vertical velocity to an explicit stratosphere model, as will be derived in section 2b.

b. Coupling to the stratosphere

c. Full, linear model

The troposphere system [Eqs. (14)–(17), (19)–(21)] is coupled to the stratosphere system [Eqs. (26)–(30)] through the two matching conditions [Eqs. (31) and (32)]. Altogether, these formulate a complete linear system, in which growing solutions whose energy decays to zero as $z^{\ast }\to \infty$ represent modes that grow in the troposphere and propagate vertically into the stratosphere.

a. Leaky modes

In this section, we focus first on solutions of the purely leaky modes, with no surface friction (F = 0). The solutions are analyzed separately: modes where υ = 0 and higher-order meridional modes where there is nonzero meridional velocity. In what follows, unless otherwise stated, the primes are dropped from the linear perturbation variables.

1) υ = 0 modes

Although the full linear model is extremely complex, restricting the solutions to υ = 0 allows for tractable analytical insight. In the troposphere, we assume solutions of the following form:

$u_0={\widehat{U}}_0(y)\exp (\mathit{ikx}+\sigma t),$(33)

where k is the zonal wavenumber, σ is the complex growth rate, and capitalized variables with hat notations are the meridional structure functions. Equivalent forms are assumed for ϕ₀, u₁, s, and s_m. Solutions in the stratosphere are assumed of the following form:

$u_s=\frac{\widehat{U_s}(y)}{\sqrt{{\rho }_0}}\exp \left[\mathit{ikx}+\sigma t+im\left(z^{*}-1\right)\right],$(34)

where m is the complex vertical wavenumber. Equivalent forms are also assumed for ϕ_s, and $w_s^{*}$. As in KE18, the meridional structure of the υ = 0 modes in the troposphere can be derived by combining Eqs. (14)–(17):

$\frac{\partial }{\partial y}\left({\widehat{S}}^{*}-{\widehat{\mathrm{\Phi }}}_0\right)=\frac{\mathit{iky}}{\sigma }\left({\widehat{S}}^{*}-{\widehat{\mathrm{\Phi }}}_0\right).$(35)

Next, we combine Eqs. (14), (16), (20), and (21) to eliminate u₀, u₁, and s_m and obtain the relationship between the meridional function of the saturation moist entropy and the barotropic geopotential:

${\widehat{S}}^{*}=\lambda {\widehat{\mathrm{\Phi }}}_0,$(36)

where

$\lambda =\frac{ik{a}_2+k^2{a}_3}{\sigma a_1+ik{a}_2+a_3{k}^2},$(37)

$a_1\equiv D(1+C)+(\chi +\sigma )(\gamma \sigma -C),$(38)

$a_2\equiv \alpha (\gamma \sigma +1),$(39)

$a_3\equiv \gamma \sigma +(G-1)C+G.$(40)

Combining Eqs. (35) and (36) gives us the meridional structure Y of the troposphere portion of the υ = 0 modes:

$Y=\exp \left(\frac{ik}{2\sigma }\hspace{0.17em}{y}^2\right),$(41)

which is equivalent in form to the meridional structure of the modes in the rigid-lid case. Furthermore, only solutions with an eastward phase speed satisfy the boundary conditions in y.

Next, we move on to solving the portion of the mode that exists in the stratosphere. With the solution form shown in Eq. (34), Eqs. (26)–(29) reduce to

$\sigma {\widehat{U}}_s+ik{\widehat{\mathrm{\Phi }}}_s=0,$(42)

$y{\widehat{U}}_s+\frac{\partial {\widehat{\mathrm{\Phi }}}_s}{\partial y}=0,$(43)

$ik{\widehat{U}}_s+\frac{m^2\sigma }{S}\hspace{0.17em}{\widehat{\mathrm{\Phi }}}_s=0,$(44)

where ${\widehat{W}}_s(y)=-\left(im\sigma /S\right){\widehat{\mathrm{\Phi }}}_s(y)$, assuming $m^2\gg {\left(H/2H_s\right)}^2$, which is a vertical shortwave approximation. Equations (42) and (44) combine into the well-known dispersion relation for the Kelvin wave:

$\sigma =\pm i\hspace{0.17em}\frac{\sqrt{S}k}{m}.$(45)

As shown in Yano and Emanuel (1991), the associated group velocity is

$c_{g,z}=\frac{k\sqrt{S}}{{\left|m\right|}^4}\left[\text{Real}{(m)}^2-\text{Imag}{(m)}^2\right],$(46)

which indicates that the vertical group velocity increases with zonal wavenumber.

Next, Eqs. (42) and (43) combine to give a meridional structure that is equivalent to that of the troposphere shown in Eq. (41). For solutions that obey the meridional boundary conditions, or that the mode amplitudes go to zero as y → ±∞, we must have that Imag(σ) < 0 (equivalent to an eastward phase speed), so we choose the positive root.

We now apply the matching conditions to derive the dispersion relation. The continuity of pressure condition demands that

${\varphi }_s\left(x,y,z^{*}=1,t\right)={\varphi }_0-V_1\left({\widehat{p}}_t\right)s^{*}.$(47)

Equations (14) and (19) combine to give the vertical velocity at the tropopause in the troposphere, which must be equal to the vertical velocity at the tropopause in the stratosphere:

$\omega \left({\widehat{p}}_t\right)=\frac{k^2}{\sigma }\hspace{0.17em}{\varphi }_0=-\frac{1}{B}\hspace{0.17em}{w}_s^{*}=\frac{1}{B}\hspace{0.17em}\frac{im\sigma }{S}\hspace{0.17em}{\widehat{\varphi }}_s,$(48)

where we note the equivalence of a rigid lid $\left[\omega \left({\widehat{p}}_t\right)=0\right]$ to the absence of the barotropic mode. Combining with Eqs. (10), (36), (45), and (47), Eq. (48) reduces to the dispersion relation:

$\underset{\text{rigid lid}}{\underbrace{\sigma a_1+a_4}}+\underset{\text{correction}}{\underbrace{\sigma {\left(kB\sqrt{S}\right)}^{-1}\left(\sigma a_1+\nu a_4\right)}}=0,$(49)

where $\nu =1-V_1\left({\widehat{p}}_t\right)$ and a₄ = ika₂ + a₃k². As shown in the underbraces, Eq. (49) is written in the form of the rigid-lid dispersion relation, plus a quartic-order correction term whose magnitude is inversely proportional to the square root of the stratosphere stratification, S. It is clear that in the limit of S → ∞, the dispersion relation reduces to that of the rigid-lid case. The quartic dispersion relation is solved numerically and the solutions are checked rigorously to satisfy the governing equations, boundary conditions, and matching conditions.

Before examining the solutions of the leaky υ = 0 modes, we first briefly derive the functional form of ω in the troposphere, in the case where υ = 0. We start by taking the time derivative of the non-dimensional continuity equation and substitute the zonal velocity using Eqs. (14) and (16):

$\sigma \hspace{0.17em}\frac{\partial \omega }{\partial \widehat{p}}=k^2\varphi .$(50)

Taking a derivative in pressure allows us to substitute in Eq. (3) and yields

$\sigma \hspace{0.17em}\frac{{\partial }^2\omega }{\partial {\widehat{p}}^2}=-k^2{\left(\frac{\partial T}{\partial \widehat{p}}\right)}_{s^{*}}{s}^{*}.$(51)

Integrating in pressure once returns

$\sigma \hspace{0.17em}\frac{\partial \omega }{\partial \widehat{p}}=-k^2{s}^{*}\overline{T}\left(\widehat{p}\right){|}_{{\mathit{s}}^{*}}+C_0,$(52)

where C₀ is an integration constant that must be determined through the boundary conditions. We integrate from the surface to the tropopause:

$\sigma \left(\omega \left({\widehat{p}}_t\right)-\omega \left({\widehat{p}}_s\right)\right)=\left({\widehat{p}}_s-{\widehat{p}}_t\right)\left(-k^2{s}^{*}\left[\overline{T}\right]+C_0\right),$(53)

which allows us to determine C₀:

$C_0=\sigma \hspace{0.17em}\frac{\omega \left({\widehat{p}}_t\right)-\omega \left({\widehat{p}}_s\right)}{{\widehat{p}}_s-{\widehat{p}}_t}+k^2{s}^{*}\left[\overline{T}\right].$(54)

Substituting for C₀ into Eq. (52), setting $\omega \left({\widehat{p}}_s\right)=0$ as earlier, and integrating once in pressure yields the non-dimensional vertical structure of ω:

$\omega \left(\widehat{p}\right)=\underset{\text{barotropic}}{\underbrace{\frac{{\widehat{p}}_s-\widehat{p}}{{\widehat{p}}_s-{\widehat{p}}_t}\hspace{0.17em}\omega \left({\widehat{p}}_t\right)}}-\underset{\text{baroclinic}}{\underbrace{\frac{k^2}{\sigma }\hspace{0.17em}{s}^{*}{\displaystyle {\int }_{{\widehat{p}}_s}^{\widehat{p}}{V}_1}\hspace{0.17em}dp}}.$(55)

As shown in the underbraces, the vertical structure of the pressure vertical velocity is a sum of the barotropic and baroclinic modes. The barotropic component changes linearly with pressure, and the baroclinic mode is zero at the surface/tropopause and peaks in the midtroposphere. The superposition of the barotropic and baroclinic mode can lead to a vertical tilt in the vertical velocity profile that depends on the phase lag between the two modes. It is worth noting that by definition, the baroclinic mode cannot interact with the stratosphere; in this model, it is only through the excitement of the barotropic mode that waves can radiate energy into the stratosphere.

Figure 1 compares the non-dimensional frequency, growth rates, phase speeds, and vertical group velocity of the leaky υ = 0 modes to those of the rigid-lid modes, with non-dimensional coefficients α = 1.5, χ = 0.5, C = 0, γ = 2, D = 0.5, G = 0.1, and S = 100. These parameters somewhat reflect Earth-like conditions in the tropics, with the exception of no cloud radiation interaction, and were specifically chosen to examine the branch of solutions that closely resemble the classical Kelvin-wave solutions but are instead amplified through the WISHE feedback. Figure 1a shows that the frequencies of the leaky waves are only slightly larger than their rigid-lid counterparts, and as such the phase speeds are slightly faster (Fig. 1c). The modification of the growth rates tell a much different story: Fig. 1b shows strong damping of the growth rate of smaller-scale waves, shifting the power spectrum toward larger scale waves, which is consistent with the results of Yano and Emanuel (1991). This reddening of the power spectrum can be physically explained through the large vertical group velocities of the smaller scale waves, as shown in Fig. 1c and indicated in Eq. (46). The smaller scale waves propagate their energy very quickly into the stratosphere, dampening their growth rate. For instance, the k = 10 wave has a vertical group velocity that nearly exceeds its zonal phase speed.

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Non-dimensional (a) frequency, (b) growth rate, and (c) phase speed and for the fastest-growing υ = 0 mode. Vertical group velocity of the leaky mode is shown in black in (c), while rigid-lid solutions are shown in blue and leaky-lid solutions shown in red. Non-dimensional parameters are α = 1.5, χ = 0.5, C = 0, γ = 2, D = 0.5, G = 0.1, and S = 100.

Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-21-0199.1

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Figure 2 shows a summary of the k = 1 eigenfunction using identical non-dimensional parameters as detailed in Fig. 1. The horizontal cross section at the surface shows a Kelvin wave pattern with surface easterlies maximizing east of the maximum temperature anomaly. Vertical velocity maximizes west of the peak temperature anomalies, and the WISHE feedback is responsible for wave growth and enhanced eastward propagation. The horizontal structure is not significantly modified from that of the rigid lid. The vertical structure of the mode exhibits a bit more complexity. Unlike the vertical structure of the rigid-lid model, which is purely baroclinic, the superposition of the barotropic and baroclinic modes leads to an eastward vertical tilt in dynamical fields, as shown in Fig. 2 and Eq. (55). Note the temperature anomalies are purely first baroclinic, since the barotropic mode is not associated with temperature perturbations The mode has a downward phase propagating, eastward tilted component in the stratosphere, which is consistent with a Kelvin-wave that has upward vertical energy propagation. As indicated by the dispersion relation, the vertical wavelength in the stratosphere is controlled by the horizontal wavenumber and stratification; a stronger stratification or shorter horizontal wavelength decreases the vertical wavelength.

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Eigenfunction of the υ = 0, k = 1 mode corresponding to the parameters detailed in Fig. 1, on (left) a horizontal cross section at the surface and (right) a vertical cross section centered on the equator. Contours indicate the saturation entropy or geopotential, where solid (dashed) lines indicate positive (negative) perturbations. Arrows indicate zonal velocity perturbations, and color shading indicates vertical velocity or ω perturbations, where indicated. Magenta line outlines the tropopause.

Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-21-0199.1

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The tropospheric system also allows for solutions of significantly slower propagating modes when cloud-radiation interaction is turned on, as shown in KE18 and E20. We select the non-dimensional parameters α = 1, χ = 1, C = 2.5, γ = 2, D = 1, G = 0.02, and S = 100, in order to obtain the slow modes as the fastest-growing solutions. Note that these parameters are slightly different from those used in E20, but still within Earth-like ranges. The use of slightly different parameters was motivated in part to compensate for the recognition that κ = 1. We also eliminated horizontal diffusion of moist static energy as compared to the model described in KE18, in order to isolate the effect of the stratosphere on the growth rates. This means that the growth rates no longer peak at low wavenumbers, contrary to what is observed, suggesting that horizontal advection may play an important role in scale-selection for slow-propagating waves. We have confirmed that the different non-dimensional parameters used and the removal of the horizontal diffusion term does not change the general structure and properties of the slow-propagating waves (not shown). Figure 3 shows the frequency, growth rate, phase speed, and vertical group velocities for the υ = 0 slow modes that can interact with the stratosphere. Aside from a small modification of the wave properties at the highest wavenumbers, these slow modes are not significantly affected by the presence of a stratosphere. Their vertical group velocities are almost negligibly small, and the corresponding vertical wavelengths in the stratosphere are extremely short. These slow modes are trapped in the troposphere and do not leak much energy into the stratosphere. Thus, these results suggest that for slow-propagating modes, such as the MJO, a rigid-lid assumption is an accurate approximation with regards to the growth rates.

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As in Fig. 1, but for the fastest-growing mode with non-dimensional parameters α = 1, χ = 1, C = 2.5, γ = 2, D = 1, G = 0.02, and S = 100.

Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-21-0199.1

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Next, we investigate the barotropic mode magnitude and phase tilt across a range of non-dimensional parameters. Combining Eqs. (14), (16), and (36) under υ = F = 0 yields the relationship between the baroclinic and barotropic zonal velocities:

$-\lambda u_0=u_1.$(56)

Here, it is evident that λ controls the phase and amplitude relationship between the barotropic and baroclinic modes. From Eq. (56), the ratio of the amplitude of the barotropic to that of the baroclinic mode is equivalent to |−λ⁻¹|, while the phase lead of the barotropic to baroclinic mode is given by tan⁻¹(λ_i/λ_r), where λ_i and λ_r are the imaginary and real components of λ, respectively. It follows that the barotropic zonal velocity to total zonal velocity ratio is |u₀/(u₀ + u₁)| = |(λ − 1)⁻¹|.

Figures 4a and 4c show the barotropic zonal velocity to total zonal velocity ratio, as well as the phase lead of the barotropic mode to the baroclinic mode, for a wide range of stratosphere stratification S. In general, we observe decreasing amplitude of the barotropic mode with increasing stratosphere stratification and zonal wavenumber. For both the classical WISHE-driven Kelvin waves (Fig. 4, top row) and radiatively destabilized slow modes (Fig. 4, bottom row), the barotropic mode is around an order of magnitude smaller than the baroclinic mode for Earth-like parameters. Furthermore, the barotropic mode leads the baroclinic mode by approximately 0.1–0.2 cycles (0.25 being a quarter cycle, or π/2). The phase lead of the barotropic mode is always positive, indicating that the eastward phase tilt in dynamical fields shown in Fig. 2 is robust across a wide-range of scales and S. In addition, an exploration of the non-dimensional parameter space did not find any solutions where the barotropic mode lagged the baroclinic mode (not shown). This result is consistent with physical intuition. For the WISHE-driven υ = 0 modes, the baroclinic easterlies peak east of the maximum temperature perturbation. If the barotropic zonal velocities are instead westward of the baroclinic zonal velocities, then the total surface easterlies could peak west of the maximum temperature anomalies, which is inconsistent with a WISHE-driven eastward-propagating mode. Note, this is inconsistent with observational analyses of convectively coupled Kelvin waves, in which a westward tilts with height in zonal wind and temperature are typically observed (Straub and Kiladis 2002). But, an explanation of the westward tilt with height in temperature is futile to search for in a strict quasi-equilibrium model, since the barotropic mode is not associated with temperature perturbations. Rather, this analysis shows that the barotropic mode, at least in WISHE-amplified Kelvin waves in which quasi-equilibrium restricts the vertical structure of buoyancy anomalies, is eastward of the first baroclinic mode. Regardless, the observed westward tilt with height in temperature may indeed indicate the importance of the second baroclinic mode over the barotropic mode for Kelvin waves (Mapes 2000; Kuang 2008a), though it is also possible to obtain the observed tilt with height in temperature in simple models where a first-baroclinic mode heating rate is imposed and convective inhibition is allowed to modulate precipitation (Raymond and Fuchs 2007). It is likely true that strict quasi-equilibrium, as employed in this study, is too restrictive an assumption for the fast-moving Kelvin waves, but relaxation of this assumption in a coupled troposphere–stratosphere model is beyond the scope of this study.

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(a),(c) Barotropic zonal velocity to total zonal velocity ratio [|u₀/(u₀ + u₁)| = |(λ − 1)⁻¹|], and (b),(d) phase lead of the barotropic mode with respect to the baroclinic mode [tan⁻¹(λ_i/λ_r)], in cycles, as functions of stratosphere stratification S and horizontal wavenumber. The plots in (a) and (b) are Generated using the non-dimensional parameters described in Fig. 1 (WISHE-modified Kelvin waves), and the plots in (c) and (d) are generated using the non-dimensional parameters described in Fig. 3 (slow υ = 0 modes). White area indicates non-growing modes.

Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-21-0199.1

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Finally, it is worth noting that the u₀ magnitude is larger than the u₁ magnitude for extremely leaky stratospheres (Fig. 4); though this may not be relevant to Earth’s atmosphere, a barotropic mode that exceeds its baroclinic counterpart leads to top-heavy vertical velocity profiles and fast propagation into the stratosphere.

2) Higher-order modes

Without the υ = 0 approximation, the solution set is extremely complex. In the rigid-lid case, the meridional structure of the eigenmodes of the full equations are parabolic cylinder functions of degree n, where n is the order of the Hermite polynomial. A brief mathematical analysis of the (intractable) leaky-lid solutions suggests that the meridional structure functions of the linear solutions are sums of parabolic cylinder functions. Note that by convention, we denote n = −1 solutions as those where υ = 0.

For these higher-order modes, we instead solve the linear problem using numerical code, which is only appropriate for finding growing modes. The troposphere system [Eqs. (14)–(17), (20)–(21)] is discretized in y, while the stratosphere system [Eqs. (26)–(29)] are discretized in y and z. Solutions are assumed to have a zonal structure of the form exp(ikx). To integrate the linear system in time, the stratosphere equations need to be transformed to a set of linear prognostic equations. The mass continuity equation [Eq. (28)] is first integrated from the lower boundary in $z^{*}$, to obtain

${\rho }_s{w}_s^{*}(y,z^{*})=w_s^{*}(y,z^{*}=1)-{\displaystyle {\int }_{z^{*}=1}^z\left\{{\rho }_s\left[ik{u}_s\left(y,z^{*}\right)+\frac{\partial }{\partial y}{\upsilon }_s\left(y,z^{*}\right)\right]\right\}}d{z}^{*},$(57)

where the tropopause velocity $w^{\ast }\left(z^{\ast }=1\right)$ is equal to the vertical velocity at the tropopause in the troposphere equations, as required from the vertical velocity matching condition. Given the vertical velocity, Eq. (29) can be used to calculate the geopotential:

$\frac{\partial }{\partial t}\hspace{0.17em}\frac{\partial }{\partial z}\varphi \left(y,z^{*}\right)=-w^{*}(y,z)S.$(58)

Integrating from the upper boundary downward gives the prognostic equation for ϕ_s:

$\frac{\partial }{\partial t}\hspace{0.17em}{\varphi }_s\left(y,z^{*}\right)=-{\displaystyle {\int }_{\infty }^z{w}_s^{*}}\left(y,z^{*}\right)S\hspace{0.17em}d{z}^{*}+C_0,$(59)

where C₀ must be determined by the upper boundary condition. An upward wave radiation boundary condition could be applied to determine C₀, as in Moskowitz and Bretherton (2000), but such a condition does not exist for the slower modes that have interactive cloud radiation and no clear analog in the stratosphere. While it is not necessary that ϕ_s = 0 as z → ∞ (as long as the energy density goes to zero), we have included a Newtonian damping in a sizeable layer extending from the upper boundary, and as such, we set C₀ = 0. Another advantage of the Newtonian damping in the upper part of the domain is to eliminate any nonphysical downward-propagating modes. For realistic values of S, however, the amplitude of the modes typically decay to zero at the top of the numerical domain. Finally, ${\varphi }_s\left(y,z^{*}=1\right)$ is connected to the troposphere through the continuity of pressure matching condition [Eq. (47)].

Spatial derivatives are approximated using fourth-order central differences, and the system is forward time-stepped using fourth-order Runge–Kutta. After specifying non-dimensional parameters, the corresponding rigid-lid solution is used to initialize the troposphere domain, while the stratosphere is initialized at rest. Since the unbalanced wave must undergo rapid gravity wave adjustment, several dampening mechanisms are used to eliminate spurious gravity wave energy. A spectral filter is applied at each time step to eliminate small-scale noise. A strong sponge-layer is included along the edges and top of the domain to eliminate reflection of gravity waves, downward-propagating waves, and spurious noise. For details on the mathematical form of the full numerical system, see appendix A. The system is integrated for a long period of time, during which the domain is periodically rescaled by a constant to prevent numerical overflow. We then isolate the growing mode of interest and infer both the complex growth rate and meridional/vertical structures. The real component of the growth rate is calculated by the linear slope of the time-varying log of the amplitude of a prognostic variable (after accounting for the rescaling). The imaginary component of the growth rate is calculated by the slope of the time-varying phase of a prognostic variable. Although we initially assume that the damping at the upper boundary does not significantly affect the stratospheric solution, the corresponding growth rate and the meridional structures of all prognostic variables are rigorously checked to satisfy the governing equations, boundary conditions, and matching conditions. As a partial test of correctness of the numerical code, the numerical solutions for the υ = 0 (n = −1) modes were cross referenced with the υ = 0 analytic solutions.

The numerical solution is robust for linear solutions that are the fastest-growing equatorially symmetric or asymmetric mode. However, for solutions that are not the fastest-growing mode, initial unbalanced energy and/or numerical error that projects onto the fastest-growing mode symmetric/asymmetric mode will cause the slower growing mode to be overtaken by the fastest-growing mode before the domain energy is concentrated solely in the mode of interest. For instance, the second-fastest-growing equatorially symmetric mode can be overtaken by the fastest-growing equatorially symmetric mode. In these cases, the meridional–vertical structure of the fastest-growing mode is isolated by integrating for a long period in time. Then, the fastest-growing mode is projected onto the domain output, and removed from the fields. If the second-fastest-growing mode does not have a weak growth rate, the remaining, filtered fields will contain the mode of interest for a long enough period of time to infer the complex growth rate and structure.

(i) WISHE classical modes

We first compare the leaky wave solutions to the rigid-lid solutions for the WISHE-destabilized classical modes of the equatorial waveguide. This is done by choosing α = 3.5, χ = 0.5, C = 0, γ = 1, D = 2.5, G = 0.25, δ_x = 15, and S = 75; note that C = 0 eliminates the slower propagating modes observed in KE18 and E20. Figure 5 shows the non-dimensional growth rates and phase speeds for select meridional orders n = −1, 0, 1, 2 leaky equatorial waves. The growth rates of the Kelvin, westward and eastward mixed Rossby–gravity, and inertia–gravity waves are clearly dampened in relation to their rigid-lid counterparts. The strength of dampening, in a percent relative sense, grows stronger with increasing wavenumber for the eastward modes. This is in contrast to the almost negligible dampening of the westward-propagating modes as |k| increases. The growth rates of the westward-propagating n = 0 and n = 1 waves are not strongly affected by the stratosphere. The phase speeds of the leaky waves are almost negligibly faster than their rigid-lid counterparts, with the largest, though still slight, modifications observed for the n = −1 solutions. The growth rates for smaller scale eastward-propagating waves are further dampened with reduced stratification in the stratosphere (cf. S = 75 and S = 25). The effect is quite pronounced; in the case shown in Fig. 5, the smallest scale rigid-lid modes possess the largest growth rates, but imposing a leaky stratosphere greatly shifts the growth rates toward larger scales. For the S = 25 case, the growth rate peaks at k = 3 for the n = −1, 0 modes, and k ≈ 5 for the n = 1, 2 modes. These results are consistent with the qualitative behavior shown in the mathematical analysis of the leaky υ = 0 modes. As such, the solutions suggest that the stratosphere acts as a reddener of the equatorial power spectrum, especially for eastward-propagating waves.

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(left) Non-dimensional growth rate for the n = −1, 0, 1, 2 modes of the rigid-lid system (solid), the leaky-lid system with S = 75 (dashed), and the leaky-lid system with S = 25 (dot–dashed). (right) Non-dimensional phase speed for the same modes, in the rigid-lid (solid) and leaky-lid system with S = 75 (black dashed). We omit k ≤ 3 for n = 2 and k ≤ 0 for n = 1 for S = 25 solutions since they do not grow rapidly enough to infer the complex growth rate. Non-dimensional parameters selected for these modes are α = 3.5, χ = 0.5, C = 0, γ = 1, D = 2.5, G = 0.25, and δ_x = 15.

Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-21-0199.1

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(ii) Slow modes

We next switch to a non-dimensional parameter set in which cloud radiative feedbacks are turned on, such that the slow modes analyzed in E20 are the fastest-growing modes. In particular, we choose α = 1, χ = 1, C = 2.5, γ = 2, D = 1, G = 0.02, δ_x = 30, and S = 100. Although the westward-propagating slow modes do grow in time under the rigid-lid model, they take too long to reach a steady state and are quickly overtaken by the eastward-propagating modes. We thus focus on the eastward-propagating slow modes, which have a much clearer analog in the real atmosphere (i.e., the MJO).

Figure 6 shows the non-dimensional growth rates and frequencies for the eastward-propagating n = −1, 1, 2 slow modes. We first compare the numerical n = −1 solutions to the analytic solutions (Fig. 6, black dots). We observe that the numerical model slightly dampens the growth rate, and only negligibly modifies the frequencies. This slight damping could be attributed to the spectral filter and/or numerical error. Assuming that the numerical error affects the wave characteristics of the n = 1, 2 slow modes the same way, we can infer that like the growth rates and frequencies of the leaky n = −1 slow modes, the growth rates and phase speeds of the higher-order slow modes are not affected by the presence of the stratosphere.

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(left) Non-dimensional growth rate for the n = −1, 1, 2, eastward-propagating slow modes of the rigid-lid system (solid) and the leaky-lid system with S = 100 (dashed) inferred from the numerical solution. (right) Non-dimensional frequency for the same modes, in the rigid-lid (solid) and leaky-lid system with S = 100 (dashed). Black dots indicate the analytic solutions of the growth rate and frequencies of the n = −1 slow modes. All n = 0 solutions are not growing. Non-dimensional parameters selected for these modes are α = 1, χ = 1, C = 2.5, γ = 2, D = 1, G = 0.02, and δ_x = 30.

Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-21-0199.1

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Perhaps of greater interest is how the slow modes interact with the stratosphere through the barotropic mode. We focus on the eastward-propagating n = 1, k = 2 mode, which has a horizontal structure that closely resembles that of the observed MJO (Emanuel 2020). Figure 7 shows the horizontal and vertical structure of the n = 1, k = 2 mode, where we observe surface cyclonic gyres westward and poleward of the area of maximum ascent. Strong westerlies are also observed westward of the maximum ascent region. The boundary layer horizontal structure very closely resembles that of the rigid-lid solution (not shown). At the tropopause (Fig. 7, top right), the horizontal structure is almost nearly the opposite of the horizontal structure at the surface, indicating the prominence of the first baroclinic mode. However, the vertical velocity at the tropopause is nonzero, and the pattern extends poleward into the cyclonic/anticyclonic gyres, an indication of the presence of the barotropic mode. If we move further up into the stratosphere, at around 20 km (Fig. 7, bottom left), the signature of the equatorial portion of the tropospheric wave disappears, and the poleward cyclonic/anticyclonic gyres become the prominent pattern. These gyres have a westward tilt with height, as indicated in the vertical cross section at y = 2.5 (Fig. 7, bottom right), though their amplitudes decay exponentially with height and are much smaller with respect to the amplitudes of the corresponding gyres in the troposphere. This indicates that the barotropic mode lags the baroclinic mode; in addition, the westward tilt height observed here is consistent with an upward-propagating Rossby wave. At the equator, the n = 1, k = 2 slow mode has a very weak eastward tilt with height (not shown), since the barotropic mode leads the baroclinic mode. Like the υ = 0 mode shown in Fig. 2, in the stratosphere, the slow mode also tilts eastward with height at the equator (not shown). This is consistent with the stratospheric eastward tilt with height of the zonal wind on the equator of observational MJO composites [see Fig. 3 of Kiladis et al. (2005)].

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Eigenfunction of the n = 1, k = 2 mode corresponding to the parameters detailed in Fig. 6, on a horizontal cross section at the (top left) the surface, (top right) tropopause (15 km), and (bottom left) 20 km. (bottom right) Vertical cross-section eigenfunction at y = 2.5. Contours indicate the geopotential, where solid (dashed) lines indicate positive (negative) perturbations. Arrows indicate wind perturbations, and color shading indicates vertical velocity perturbations at the level indicated, except for the boundary layer cross section, where color shading indicates midlevel vertical velocity.

Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-21-0199.1

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b. Frictionally modified modes

From Eq. (60), we see that a barotropic mode at rest cannot be excited if there is no surface friction (F = 0). Thus, surface friction acts to couple the barotropic mode with the baroclinic mode; since the forcing is primarily in the first baroclinic mode; it is the first baroclinic mode that excites the barotropic mode.

The barotropic geopotential ϕ₀ is solved by inverting Eq. (60). Equations (14)–(17), (20), and (21) form the complete linear system. The system is solved numerically by discretizing in y. Spatial derivatives are again approximated using fourth-order central differences, and the system is forward time-stepped using fourth-order Runge–Kutta. As before, the frictionless, rigid-lid solution is used to initialize the domain, after which we integrate the system for a long period of time and then isolate the growing mode of interest.

Before proceeding, it is important to note that the n = −1 modes in this section differ from the traditional definition of υ = 0, in that the meridional wind is allowed to be nonzero but small. This is deliberately chosen, since υ₀ = υ₁ = 0 solutions under zero vertical velocity boundary conditions require the barotropic mode to be divergence free; this is only achievable for nonzero u₀ if there are also nonzero υ₀. While solutions of υ₀ = υ₁ = 0 under the aforementioned conditions do exist, they are just classical frictionally damped baroclinic mode solutions. However, we are interested in the solutions in which the barotropic mode can be excited, and as such, allow υ to be small for the n = −1 modes. We investigate the complex growth rates as a function of the non-dimensional surface friction, F, for which an Earth-like range is F ≈ 0.1–0.4. Figure 8 shows the growth rate of the n = −1, n = 0, and n = 1 modes for the same parameter choices as in Fig. 5, which selects for WISHE-modified Matsuno modes. Figure 8 indicates that the surface friction acts as a damping effect on all wavelengths, and the strength of the damping is nearly constant across all wavelengths. This is expected as the aerodynamic drag law acts on a fixed damping time scale. The phase speeds are not significantly changed from the phase speeds of the rigid-lid modes. The results are consistent with our formulation of surface friction, in that surface friction does not “force convection,” as convection is assumed to be in a strict, statistical equilibrium.

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(left) Non-dimensional growth rate for the n = −1, 0, 1 modes of the rigid-lid system (solid), the surface friction + rigid-lid system with F = 0.1 (dashed), and the surface friction + rigid-lid system F = 0.25 (dot–dashed). The n = −1 modes differ from the traditional definition of υ = 0, in that the meridional wind is allowed to be nonzero but small. (right) Non-dimensional phase speed for the same modes, in the rigid-lid (solid) and surface friction + rigid-lid system with F = 0.1 (black dashed). All n = 2 and the n = 1 for k ≤ 0 solutions are omitted since they do not grow rapidly enough to infer the complex growth rate. Non-dimensional parameters selected for these modes are α = 3.5, χ = 0.5, C = 0, γ = 1, D = 2.5, G = 0.25, and δ_x = 15.

Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-21-0199.1

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As illustrated in Eq. (60), surface friction can lead to an excitation of the barotropic mode, even under a rigid lid. Figures 9a and 9b show the meridional structure of the n = −1, k = 1 barotropic zonal and meridional velocities as compared to their baroclinic counterparts. Like the leaky-lid case, the barotropic zonal wind magnitude is around an order of magnitude smaller than the baroclinic mode. However, while the baroclinic mode magnitudes are primarily confined close to the equator (around y = −5 to y = 5), the barotropic mode wind velocities have long tails that extend far away from the equator. To understand this effect, at least on a high level, we consider neutral barotropic and baroclinic equatorial waves. Since a neutral barotropic wave will have a larger phase speed (and equivalent depth) than a neutral baroclinic wave, it will consequently have a larger meridional length scale and a more poleward turning latitude [see Eq. (8.77) in Vallis (2017)]. This qualitatively explains the increased poleward extent of the barotropic mode as compared to the baroclinic mode. A more quantitative discussion, under the simplest possible dynamics, is discussed in appendix B. The barotropic mode might be an important teleconnection mechanism between the tropics and the extratropics, but further analysis of this effect is out of the scope of this study (Horel and Wallace 1981). An important caveat to note, however, is that a rigid-lid barotropic mode is completely trapped in the troposphere; the addition of a leaky stratosphere may limit the poleward extent of the barotropic mode, as will be examined in the next section.

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(a) Meridional structure of 15 times the barotropic zonal wind and the baroclinic zonal wind for the n = −1, k = 1 mode with F = 0.25 and the parameter set chosen in Fig. 8. (b) As in (a), but for the barotropic and baroclinic meridional wind. (c) Horizontal structure of the boundary layer barotropic vorticity (shading) and baroclinic vorticity (contours). (d) Equatorial vertical cross sections of the pressure vertical velocity (shading) and zonal wind perturbations (contours) for the same mode. Solid (dashed) contours are positive (negative) zonal wind anomalies.

Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-21-0199.1

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Under a rigid lid the nondivergent barotropic mode cannot be associated with vertical velocity perturbations, though a vertical tilt can still exist in the horizontal wind fields from the superposition of the barotropic and baroclinic horizontal winds. Figure 9d shows the vertical cross section on the equator of the n = −1, k = 1 mode. The vertical structure of the vertical velocity is only first baroclinic, despite the zonal wind exhibiting a slight eastward vertical tilt. This eastward vertical tilt in the zonal wind field is consistent with the eastward tilt observed in the purely leaky n = −1 modes. This is because the barotropic mode leads the baroclinic mode, as can also be seen in the horizontal structure of the boundary layer vorticity decomposition in Fig. 9c.

Unfortunately the numerical model is quite unstable for the slow modes in the rigid-lid, surface friction limit. Despite strong sponge layers, instabilities unrelated to the slow modes (spurious noise, gravity waves) develop quickly in the numerical domain, precluding inferral of the complex growth rate and horizontal structure. The behavior of the barotropic mode in the slow modes will be discussed in the following section using the fully coupled, surface friction model, albeit with a large stratosphere stratification, instead of a rigid lid.

These results show that surface friction act strictly as a damping mechanism in our framework, contrary to the results shown in Moskowitz and Bretherton (2000). This is likely because the convective heating associated with frictional convergence cannot “force” the system, as convection must be balanced by adiabatic cooling from large-scale subsidence and downdrafts that import low entropy midlevel air into the boundary layer. Surface friction does, however, act to modify both the horizontal and vertical structure of the equatorial waves, as shown through the long poleward tails of the barotropic mode and vertical tilt in the zonal wind fields.

c. Leaky, frictionally modified modes

While examining the excitation of the barotropic mode in the limits of surface friction under a rigid lid and a frictionless surface under a leaky lid were both useful exercises to isolate their respective effects on tropical waves, in the real world, both surface friction and leakage of energy to the stratosphere can act simultaneously to modify equatorial waves through excitation of the barotropic mode. In particular, the long tails of the tropospherically trapped barotropic mode observed in the rigid lid, surface friction model may behave differently if the barotropic mode can leak energy into the stratosphere.

To understand the extent to which both of these mechanisms can interact, we run the full troposphere–stratosphere numerical model with nonzero surface friction, using the same methods to infer the complex growth rates and eigenmodes. We first examine their combined effect on the WISHE-driven classical Matsuno modes through the same parameter set as used in Figs. 5 and 8. In addition, we focus on the eastward-propagating modes since the westward-propagating modes were not strongly affected by the presence of the stratosphere.

Figures 10a and 10b show the non-dimensional growth rate and phase speeds for the WISHE destabilized Matsuno modes, respectively. Since there is both energy leakage into the stratosphere and surface friction, the growth rates are strongly dampened from the equivalent modes in the rigid-lid, inviscid limit. The frequencies of the WISHE Matsuno modes are not greatly modified, though all of the n = 0 modes and smaller scale n = −1 modes (k ≥ 5) have slower phase speeds than their rigid-lid counterparts. Comparing with the growth rates in the stratosphere-only case shown in Fig. 5, we can see that the damping effects of the stratosphere and surface friction are approximately additive.

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(a),(c) Non-dimensional growth rate and (b),(d) non-dimensional phase speed for the n = −1, 0, 1 modes of the rigid-lid system (solid), the surface friction + rigid-lid system with S = 150 and F = 0.1 (dashed), and the surface friction + rigid-lid system with S = 75 and F = 0.1 (dot–dashed). For (a) and (b), the non-dimensional parameters selected are α = 3.5, χ = 0.5, C = 0, γ = 1, D = 2.5, G = 0.25, and δ_x = 15; for (c) and (d), these are α = 1, χ = 1, C = 2.5, γ = 2, D = 1, G = 0.02, and δ_x = 30. Only S = 75 case is shown for (c) and (d).

Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-21-0199.1

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The more interesting question, perhaps, is if the barotropic mode behaves differently when it can be excited by surface friction and also interact with the stratosphere. To examine this, we decompose the horizontal divergence of the barotropic mode into $\partial u/\partial x$, $\partial \upsilon /\partial y$, and the sum of both, which is equivalent to the pressure vertical velocity at the tropopause. Figure 11 shows the barotropic mode velocities and the barotropic horizontal divergence decomposition for the n = 0, k = 1 WISHE Matsuno mode, for a realistic stratosphere stratification (S = 75), as well as a highly stratified one (S = 500). In the case with a highly stratified stratosphere, we see the long-tail feature of the barotropic mode velocities, as also seen in the surface friction under a rigid-lid model. However, a key differing feature is that the barotropic mode velocities are no longer completely nondivergent, as evidenced in Fig. 11b. Further, the horizontal divergence of the barotropic mode (or, equivalently, the tropopause vertical velocity) decays to zero very quickly poleward, around |y| = 3. This is because near the equator, $\partial u_0/\partial x$ and $\partial {\upsilon }_0/\partial y$ have the same sign, evidence that the barotropic mode is exciting the stratosphere near the equator. In contrast, despite the barotropic velocities being small (though nonzero) poleward of the equator, the horizontal divergence is zero because $\partial u_0/\partial x$ and $\partial {\upsilon }_0/\partial y$ have opposite signs and almost exactly cancel. This is evidence of a tropospherically trapped barotropic mode. When we reduce the stratosphere stratification to S = 75, as shown in Figs. 11c and 11d, in effect allowing the tropopause to be leakier, we see that the poleward extent of the barotropic mode velocities is greatly reduced, and hence the poleward extent of $\partial u_0/\partial x$ and $\partial {\upsilon }_0/\partial y$ is also reduced. The horizontal divergence still decays to zero around |y| = 3, as in the case where S = 500.

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(a),(c) Barotropic mode zonal and meridional velocities for the WISHE modified Matsuno n = 0, k = 1 mode using the same parameter set as in Fig. 8, that is, α = 3.5, χ = 0.5, C = 0, γ = 1, D = 2.5, G = 0.25, δ_x = 15, and F = 0.1, but with (a) S = 500 and (c) S = 75. (b),(d) $2\left(\partial u_0/\partial x\right)$ and $2\left(\partial {\upsilon }_0/\partial y\right)$ (scaled by 2), and $\mathbf{\nabla }\cdot v_0$ for the same mode, but with (b) S = 500 and (d) S = 75.

Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-21-0199.1

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These numerical experiments illustrate the role the S plays in modulating the poleward extent of the barotropic mode. The mechanism can be reasoned as the following: under infinite stratification, (the rigid lid), surface friction allows for the excitement of the barotropic mode far away from the equator, as shown earlier. As the stratification of the stratosphere is decreased, more wave energy escapes into the stratosphere, reducing the poleward extent of the barotropic mode. Despite the unrealistically large stratification (S = 500) used to emphasize the characteristics of the trapped barotropic mode, the qualitative behavior of the model does not change with intermediate stratifications (not shown). In other words, the poleward extent of the tropospherically trapped barotropic mode increases with S.

Figures 10c and 10d show the non-dimensional growth rate and phase speeds for the slow modes, for which we obtain by choosing the same parameter set as used in Fig. 3 and 6. Note, the n = 0 waves do not grow in time. The growth rates are not modified greatly from the rigid-lid solutions; surface friction seems to have a weaker damping effect on the slower propagating modes. The damping effect of the stratosphere on the growth rates is small but the greatest at the largest scales, and almost negligible for the smaller scale waves, as before. However, the frequencies/phase speeds for the slow modes are reduced, by as much as 30% for the k = 1 wave. The damping of the frequency is much greater for the n = 1 slow propagating modes than the n = −1 modes, and much stronger for larger scale n = 1 waves. It is also worth noting that the S = 150 case is not shown for the slow modes since the lines are nearly indistinguishable from the S = 75 case.

The behavior of the poleward extent of the barotropic mode for the slow modes is similar to that for the WISHE-modified Matsuno modes. Figure 12 shows that the barotropic mode associated with the n = 1, k = 2 slow mode becomes nondivergent away from the equator and trapped in the troposphere. However, one key difference is that the slow modes do not leak much energy into the stratosphere to begin with, and thus decreasing the stratosphere stratification does not completely eliminate the long-tailed behavior of the barotropic velocities. This is evidenced by the small magnitude of the horizontal divergence of the barotropic mode (Figs. 12b,d), regardless of the stratosphere stratification, as iku₀ and $\partial {\upsilon }_0/\partial y$ nearly cancel close to the equator (|y| < 3), and exactly cancel for |y| > 3.

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As in Fig. 11 but for the slow-propagating n = 1, k = 2 mode using the parameter set α = 1, χ = 1, C = 2.5, γ = 2, D = 1, G = 0.02, δ_x = 30, and F = 0.1.

Citation: Journal of the Atmospheric Sciences 79, 3; 10.1175/JAS-D-21-0199.1

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What are the physical parameters that control the non-dimensional stratosphere stratification, S, and how can they vary across different equatorial waves? The non-dimensionalization of S [see appendix A] suggests that the stratosphere buoyancy frequency N² and the meridional length scale L_y are quantities that could potentially lead to large variations in S. The value of L_y depends on a multitude of factors that can vary greatly in the tropics, such as the troposphere dry stratification, moist adiabatic lapse rate, and the precipitation efficiency. The largest influencing factor, however, is the precipitation efficiency ϵ_p: waves with larger ϵ_p experience a much greater stratosphere stratification. For instance, for Earth-like parameters and a ϵ_p = 0.5, S ≈ 20, while for ϵ_p = 0.95, S ≈ 200, an order of magnitude difference. While our definition of ϵ_p is an egregiously simple parameterization of cloud microphysics, the impact of ϵ_p on S links cloud microphysical properties to the behavior of the barotropic mode.