Editorial Type:
Article
Article Type:
Research Article

Diurnally Forced Tropical Gravity Waves under Varying Stability

Ewan Short aSchool of Geography, Earth and Atmospheric Sciences, University of Melbourne, Melbourne, Victoria, Australia
bARC Centre of Excellence for Climate Extremes, University of Melbourne, Melbourne, Victoria, Australia

Search for other papers by Ewan Short in
Current site
Google Scholar
PubMed Close
https://orcid.org/0000-0003-2821-8151
,
Todd P. Lane aSchool of Geography, Earth and Atmospheric Sciences, University of Melbourne, Melbourne, Victoria, Australia
bARC Centre of Excellence for Climate Extremes, University of Melbourne, Melbourne, Victoria, Australia

Search for other papers by Todd P. Lane in
Current site
Google Scholar
PubMed Close
,
Craig H. Bishop aSchool of Geography, Earth and Atmospheric Sciences, University of Melbourne, Melbourne, Victoria, Australia
bARC Centre of Excellence for Climate Extremes, University of Melbourne, Melbourne, Victoria, Australia

Search for other papers by Craig H. Bishop in
Current site
Google Scholar
PubMed Close
, and
Matthew C. Wheeler cBureau of Meteorology, Melbourne, Victoria, Australia

Search for other papers by Matthew C. Wheeler in
Current site
Google Scholar
PubMed Close
Online Publication:
20 Oct 2023
Print Publication:
01 Oct 2023
Collections:
Years of the Maritime Continent
DOI:
https://doi.org/10.1175/JAS-D-23-0074.1
Page(s):
2557–2579
Free access

Abstract

Diurnal processes play a primary role in tropical weather. A leading hypothesis is that atmospheric gravity waves diurnally forced near coastlines propagate both offshore and inland, encouraging convection as they do so. In this study we extend the linear analytic theory of diurnally forced gravity waves, allowing for discontinuities in stability and for linear changes in stability over a finite-depth “transition layer.” As an illustrative example, we first consider the response to a commonly studied heating function emulating diurnally oscillating coastal temperature gradients, with a low-level stability change between the boundary layer and troposphere. Gravity wave rays resembling the upper branches of “Saint Andrew’s cross” are forced along the coastline at the surface, with the stability changes inducing reflection, refraction, and ducting of the individual waves comprising the rays, with analogous behavior evident in the rays themselves. Refraction occurs smoothly in the transition-layer solution, with substantially less reflection than in the discontinuous solution. Second, we consider a new heating function which emulates an upper-level convective heating diurnal cycle, and consider stability changes associated with the tropical tropopause. Reflection, refraction, and ducting again occur, with the lower branches of Saint Andrew’s cross now evident. We compare these solutions to observations taken during the Years of the Maritime Continent field campaign, noting better qualitative agreement with the transition-layer solution than the discontinuous solution, suggesting the tropopause is an even weaker gravity wave reflector than previously thought.

Significance Statement

This study extends our theoretical understanding of how forced atmospheric gravity waves change with atmospheric structure. Gravity wave behavior depends on atmospheric stability: how much the atmosphere resists vertical displacements of air. Where stability changes, waves reflect and refract, analogously to when light passes from water to air. Our study presents new mathematical tools for understanding this reflection and refraction, demonstrating reflection is substantially weaker when stability increases over “transition layers,” than when stability increases suddenly. Our results suggest the tropical tropopause reflects less gravity wave energy than previously thought, with potential design implications for weather and climate models, to be assessed in future work.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

This article is included in the Years of the Maritime Continent Special Collection.

Corresponding author: Ewan Short, shorte1@student.unimelb.edu.au

Abstract

Diurnal processes play a primary role in tropical weather. A leading hypothesis is that atmospheric gravity waves diurnally forced near coastlines propagate both offshore and inland, encouraging convection as they do so. In this study we extend the linear analytic theory of diurnally forced gravity waves, allowing for discontinuities in stability and for linear changes in stability over a finite-depth “transition layer.” As an illustrative example, we first consider the response to a commonly studied heating function emulating diurnally oscillating coastal temperature gradients, with a low-level stability change between the boundary layer and troposphere. Gravity wave rays resembling the upper branches of “Saint Andrew’s cross” are forced along the coastline at the surface, with the stability changes inducing reflection, refraction, and ducting of the individual waves comprising the rays, with analogous behavior evident in the rays themselves. Refraction occurs smoothly in the transition-layer solution, with substantially less reflection than in the discontinuous solution. Second, we consider a new heating function which emulates an upper-level convective heating diurnal cycle, and consider stability changes associated with the tropical tropopause. Reflection, refraction, and ducting again occur, with the lower branches of Saint Andrew’s cross now evident. We compare these solutions to observations taken during the Years of the Maritime Continent field campaign, noting better qualitative agreement with the transition-layer solution than the discontinuous solution, suggesting the tropopause is an even weaker gravity wave reflector than previously thought.

Significance Statement

This study extends our theoretical understanding of how forced atmospheric gravity waves change with atmospheric structure. Gravity wave behavior depends on atmospheric stability: how much the atmosphere resists vertical displacements of air. Where stability changes, waves reflect and refract, analogously to when light passes from water to air. Our study presents new mathematical tools for understanding this reflection and refraction, demonstrating reflection is substantially weaker when stability increases over “transition layers,” than when stability increases suddenly. Our results suggest the tropical tropopause reflects less gravity wave energy than previously thought, with potential design implications for weather and climate models, to be assessed in future work.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

This article is included in the Years of the Maritime Continent Special Collection.

Corresponding author: Ewan Short, shorte1@student.unimelb.edu.au
Keywords: Tropics; Inertia-gravity waves; Internal waves; Tropopause; Differential equations; Fourier analysis

1. Introduction

Diurnal processes determine much of the variability of tropical weather. For rainfall, the amplitude of the mean diurnal harmonic exceeds 70% of the seasonal mean in many coastal tropical regions (Minobe and Takebayashi 2015). In locations like the Maritime Continent and the Bight of Panama, coastal processes themselves likely induce between 40% and 60% of rainfall (Bergemann et al. 2015). Satellite observations indicate that the peak of the rainfall diurnal harmonic occurs progressively later with distance from the coastline, in some locations propagating up to 1000 km offshore (e.g., Yang and Slingo 2001). Analogous behavior has been observed for the diurnal cycle of winds (Gille et al. 2005), leading to many hypotheses on the interconnectedness of these processes (e.g., Mori et al. 2004; Robinson et al. 2008; Vincent and Lane 2016; Kilpatrick et al. 2017).

One theory, originating with Mapes et al. (2003), is that gravity waves forced along or near coastlines propagate both offshore and inland, destabilizing the atmosphere and promoting convection as they go. These waves may be forced by topography or convective heating near coastlines, or by the diurnally oscillating temperature gradient associated with the coastline itself. In the tropics, gravity waves forced by coastal temperature gradients can be interpreted as the linear component of the land–sea breeze, manifesting as onshore surface winds in the afternoon and evening (the sea breeze), and offshore winds in the morning (the land breeze).

Rotunno (1983) demonstrated using a simple, two-dimensional linear theory that for latitudes between +30° and −30°, oscillating coastal temperature gradients force gravity wave rays, i.e., packets of gravity waves which propagate far offshore. The rays resemble the upper half of “Saint Andrew’s cross,” the signature of internal gravity waves generated by oscillatory forcings (Mowbray and Rarity 1967). The differing behavior of the land–sea breeze poleward and equatorward of 30° was subsequently corroborated by nonlinear numerical simulations (Yan and Anthes 1987).

Rotunno’s original theory employed the Boussinesq equations, linearized about a hydrostatic, stationary atmosphere, with a free-slip lower boundary. Subsequent studies modified his approach to incorporate momentum diffusivity (Niino 1987), friction (Dalu and Pielke 1989; Du and Rotunno 2015; Das Gupta et al. 2015), shore-parallel thermal wind shear (Drobinski et al. 2011), nonlinear coastlines (Li and Chao 2016; Jiang 2012a), constant background winds (Qian et al. 2009; Du and Rotunno 2018), and background winds that shear linearly with height (Du et al. 2019). Of particular relevance to the present study is the theory of Jiang (2012b), who divided the vertical domain into at most 3 distinct layers, allowing discontinuities in the Brunt–Väisälä frequency N, potential temperature, and background winds between each layer, but requiring they be constant within each layer. To solve the resulting equations, Jiang modified the heating function introduced by Rotunno to constrain heating to lie within a single layer.

In the present study, we consider some new extensions of Rotunno’s theory to situations involving nonconstant stability, i.e., a nonconstant Brunt–Väisälä frequency N. We first consider the case of a step change in N, i.e., N = N1 below some height H1 and N = N2 above H1, where N1 and N2 are distinct positive numbers. This analysis is similar to that of Lindzen and Tung (1976), who considered the anelastic equations, and calculated the reflection and refraction coefficients associated with a stability discontinuity for a single gravity wave forced below the stability change. Here we consider the simpler Boussinesq equations, but the more complex case of a forced spectrum of waves, and the effect of forcing above the stability change. This analysis is also similar to that of Jiang (2012b) discussed above, although here we consider the simpler problem of resting horizontal background winds, but the more complex situation of forcing above and below the stability change, and the possibility of a stability decrease with height.

Second, we consider the case where N = N1 below some height H1, N = N2 above another height H2, with H2 > H1, and with N transitioning linearly between N1 and N2 between H1 and H2. This setup admits an analytic solution that generalizes the piecewise-constant solution. To the best of our knowledge, a solution to this problem is yet to appear in the published literature, with the closest solution being for the three-layer model considered by Jiang (2012b), in which waves are forced only in the lowest vertical layer, and N is constant within each layer. We will refer to our two solutions as the “piecewise-constant” and “transition-layer” solutions, respectively. We present illustrative examples of both solutions with a low-level stability change emblematic of that between the boundary layer and free troposphere, as this setup is easiest to interpret and compare with previous work.

We then generalize the theory by considering an alternative heating function which emulates the upper-level heating associated with a diurnally recurring convective line. This extension is straightforward, as our core mathematical results are independent of the spatial structure of the heating function. We use this extended theory to interpret soundings acquired during the Australian leg of the Years of the Maritime Continent (YMC) campaign (Yoneyama and Zhang 2020), focusing on the stability change between the troposphere and stratosphere.

The remainder of this paper is organized as follows. In sections 2 and 3 we present the piecewise-constant and transition-layer solutions, respectively, and provide illustrative example solutions using the original low-level surface heating function of Rotunno (1983). In section 4 we introduce the YMC sounding data and the alternative upper-level convective diurnal heating function, and interpret the YMC data in light of the extended theory. Section 5 provides discussion and conclusions.

2. Piecewise-constant solution

We begin with the dimensional equations for Boussinesq, hydrostatic flow, linearized about a resting background state:
∂u*∂t*−fυ*=−∂p*∂x*,
∂υ*∂t*+fu*=0,
∂p*∂z*=b*,
∂b*∂t*+N(z)2w*=Q*,
∂u*∂x*+∂w*∂z*=0,
Q*=Q0π(π2+tan⁡−1x*L)exp⁡(−z*H)cos⁡(ωt*),
w*(z*=0)=0,
where x* is the shore-perpendicular direction, with x*<0 and x*>0 representing sea and land, respectively. The time variable t* denotes seconds since noon local solar time (LST). The Coriolis parameter f = 2ω sinϕ is assumed constant, ω the angular frequency of Earth, and ϕ the latitude, with |ϕ| < 30°. The functions u*, υ*, and w* are the velocity perturbations, p* is the Boussinesq disturbance pressure, and b*=gθ′/θ0 is the buoyancy, where g is gravity, and θ0 and θ′ are the surface and deviation potential temperatures, respectively. The function Q* represents the coastal heating cycle, with L the coastal width, Q0 the heating amplitude, and H the vertical scale of the heating. As discussed by Rotunno (1983), this function parameterizes surface solar heating, and the turbulent diffusion of thermal energy through the lower atmosphere, idealizing this diffusion as instantaneous. Figure 1 illustrates Q* for Q0 = 1.2 × 10−5 m s−3, L = 50 km, and H = 1 km.
Fig. 1.
Fig. 1.

The heating function given by Eq. (6) at 1200 LST, i.e., t*=0, with L = 50 km, H = 1 km, and Q0 = 1.2 × 10−5 m s−3.

Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-23-0074.1

Gravity wave solutions to Eqs. (1)–(7) are, strictly speaking, two-dimensional linear Boussinesq hydrostatic internal gravity waves; we will subsequently refer to these simply as “waves.” Equations (1)–(7) are identical to those considered by Rotunno (1983), except we no longer assume N is constant. Instead, we take
N(z)={N1,0≤z*≤H1,N2,z*>H1,
where N1 and N2 are positive constants, and H1 > 0. Following Qian et al. (2009), we define the nondimensional variables
x*=N1Hωx,y*=N1Hωy,z*=Hz,t*=tω,Q*=Q0Q,u*=Q0N1ωu,υ*=Q0N1ωυ,w*=Q0N12w,b*=Q0ωb,p*=Q0Hωp,
noting t = 2π corresponds to t*=1 day in dimensional coordinates. Substituting, we obtain a set of dimensionless equations,
ut−fωυ=−px,
υt+fωu=0,
pz=b,
ux+wz=0,
bt+n(z)2w=Q,
Q=1π(π2+tan⁡−1xL)exp⁡(−z)cos⁡(t),
w(z=0)=0,
where L=ωL/(N1H) is the nondimensional coastal width. We define additional nondimensional parameters H1=H1/H and N=N2/N1, where N≠1, and the vertical subdomains D1=[0,H1] and D2=(H1,∞). The function n(z) is then given by
n(z)={1,z∈D1,N,z∈D2.
Defining the stream function ψ by (u, w) = (ψz, −ψx), Eqs. (10)–(16) can be combined to yield
ψzztt+f2ω2ψzz+n(z)2ψxx=−Qx.
Taking the Fourier transform with respect to x of both sides of Eq. (18) gives
ψ˜zztt+f2ω2ψ˜zz−κ2n(z)2ψ˜=−e−L|κ|e−zeit+e−it2,
where ψ˜(κ,z,t)=F[ψ(x,z,t)]=∫−∞∞ψe−iκxdx is the Fourier transform of ψ. As noted by Qian et al. (2009), the inverse Fourier transform can be calculated as
ψ=F−1[ψ˜(κ,z,t)]=Re[1π∫0∞ψ˜(κ,z,t)eiκxdκ],
so we can assume without loss of generality κ > 0 in Eq. (19). Linear independence of eit and e−it implies we may express ψ˜(κ,z,t)=ψ^1(κ,z)eit+ψ^2(κ,z)e−it and solve for ψ^1 and ψ^2 separately, which, as noted by Qian et al. (2009), contribute the leftward- and rightward-propagating components of ψ, respectively. The perturbation velocities u and w are then
u=F−1(∂ψ˜∂z),
w=F−1(iκψ˜).
Consider first the positive mode ψ^1, dropping the subscript while the context is clear. Equation (19) reduces to
ψ^zz+m2n(z)2ψ^=12A2e−Lκe−z,
where we have defined the nondimensional number A=1−f2/ω2, which is real for latitudes between +30° and −30°, and m=κ/A>0, which is the nondimensional vertical wavenumber for z ∈ D1, noting the corresponding dimensional vertical wavenumber is m*=m/H. As discussed by Rotunno (1983), A determines the aspect ratio of the nondimensional solution, which becomes unphysical as f → ω due to the absence of friction from the theory. As noted by Jiang (2012b), conservation of mass requires continuity of w, and therefore of ψ^, at H1. Furthermore, as noted by Lindzen and Tung (1976), integrating Eq. (23) over the interval (H1−ϵ,H1+ϵ), and taking the limit as ϵ → 0, we obtain the condition that ψ^z, and therefore u, must also be continuous at H1. Note the above reductions and matching conditions also apply to ψ^2.
Before solving Eq. (23), consider first its unforced analog,
ψ^zz+m2n(z)2ψ^=0.
This reduces to a pair of wave equations for z ∈ D1 and z ∈ D2. The general solutions to these equations are given by linear combinations of eim(z−H1) and e−im(z−H1) for z ∈ D1, and linear combinations of eimN(z−H1) and e−imN(z−H1) for z ∈ D2. To develop physical intuition, we first consider two simpler subproblems.
First, ignore the lower boundary, but apply the matching conditions at H1, and the radiation condition above H1. Recall that gravity waves propagate energy at the group velocity, the vertical component of which is negative that of the phase velocity. Thus, for the positive mode eit, the radiation condition requires we select only the eimN(z−H1) mode above H1. The general solution thus reduces to
ψ^={CN+12eim(z−H1)+C1−N2e−im(z−H1),z∈D1,CeimN(z−H1),z∈D2,
where C∈C is an unknown constant. This solution represents a wave propagating energy upward, that is partially reflected and partially refracted at H1. By taking ratios of amplitudes, we can define the reflection and refraction coefficients
r1=|1−N1+N|,
r2=2N+1,
respectively, which are special cases of those derived by Lindzen and Tung (1976). As N→∞, there is perfect reflection, and no refraction, with z=H1 acting like a rigid upper boundary. As N→0, we also approach perfect reflection, but the incident and reflected waves are now in phase at H1, implying the amplitude of the refracted wave must be twice that of the incident wave.
Second, suppose we solve Eq. (24) again, reintroducing the rigid lower boundary at z = 0, but not applying the radiation condition. We again apply the matching conditions at H1. The general solution is now
ψ^={B2ieimz−B2ie−imz,z∈D1,B2iNReimN(z−H1)−B2iNR¯e−imN(z−H1),z∈D2,
where B∈C is an unknown constant, and R=cos⁡(mH1)+iNsin⁡(mH1), with R¯ the complex conjugate of R. By writing R in complex polar form, Eq. (28) may also be expressed,
ψ^={Bsin⁡(mz),z∈D1,Bcos⁡2(mH1)+N2sin⁡2(mH1)Nsin⁡[mN(z−H1)+Arg(R)],z∈D2,
where Arg(R) denotes the complex argument of R. Equation (28) describes the vertical structure of the wave that results from the interference of the upward- and downward-propagating disturbances, noting the resulting structure does not propagate in the vertical. We define the “ducting coefficient” r3 by taking the ratio of the amplitudes of the waves in D1 and D2:
r3=Ncos⁡2(mH1)+N2sin⁡2(mH1).
It follows that r3 has stationary points when mH1=π(2k−1)/2 and mH1=πk, where k is a natural number, from which resonance conditions can be inferred. Note that as N→0, r3 → 0, and z=H1 behaves like a rigid lower boundary. As N→∞,r3→|1/sin⁡(mH1)|, noting r3 → ∞ as mH1→kπ, as from Eq. (28), the amplitude of the wave in D2 approaches zero. Figures showing example waves corresponding to Eqs. (25) and (28) are provided in the online supplement.
Now consider the forced equation, Eq. (23), which, following Rotunno (1983), may be solved using Green’s method. We first solve for the function G(κ, z), which gives the response to a point forcing at z′,
Gzz+m2n(z)2G=δ(z−z′)
for arbitrary z′, where δ is the Dirac delta function, with ψ^ then given by
ψ^(z)=12A2e−Lκ∫0∞G(z,z′)e−z′ dz′.
We require G be continuous at z′, and by integrating Eq. (31) over (z′ − ϵ, z′ + ϵ) and taking the limit as ϵ → 0 we obtain the condition
lim⁡ϵ→0[∂G∂z(κ,z′+ϵ)−∂G∂z(κ,z′−ϵ)]=1.
When z′ ∈ D1, waves forced above z′ partially reflect and refract at H1, as in Eq. (25). Waves forced below z′ perfectly reflect off the rigid lower boundary, resulting in a vertically nonpropagating wave below z′. The matching conditions at z′ then allow the constant C in Eq. (25), and the amplitude of the vertically nonpropagating wave below z′, to be solved for in terms of z′. Thus, for z′ ∈ D1 we obtain
G={−1mP[N+12eim(z′−H1)+1−N2e−im(z′−H1)]sin⁡(mz),z∈D1,z≤z′,(34a)−1mPsin⁡(mz′)[N+12eim(z−H1)+1−N2e−im(z−H1)],z∈D1,z′<z,(34b)−1mPsin⁡(mz′)eimN(z−H1),z∈D2,(34c)
where
P=1+N2e−imH1+1−N2eimH1.
When z′ ∈ D2, a vertically nonpropagating wave is forced below z′, with partial ducting within D1, as in Eq. (28). Above z′ we apply the radiation condition. The matching conditions at z′ then allow the constant B in Eq. (28), and the amplitude of the wave above z′, to be solved for in terms of z′. Thus, for z′ ∈ D2 we have
G={−1mR¯eimN(z′−H1)sin⁡(mz),z∈D1,(36a)−12imNR¯eimN(z′−H1)[ReimN(z−H1)−R¯e−imN(z−H1)],z∈D2,z≤z′,(36b)−12imNR¯[ReimN(z′−H1)−R¯e−imN(z′−H1)]eimN(z−H1),z∈D2,z′<z,(36c)
where R=cos⁡(mH1)+iNsin⁡(mH1) as before. Figures showing waves corresponding to the functions G(κ, z) are provided in the online supplement.
Equation (32) can be calculated by decomposing the integral into two terms F1 and F2, involving forcing below and above H1, respectively. Explicitly, we have ψ^=F1+F2, where
F1=12A2e−Lκ∫0H1G(z,z′)e−z′dz′,
F2=12A2e−Lκ∫H1∞G(z,z′)e−z′dz′.
These integrals can be calculated analytically. For instance, if z ∈ D1, the term F1 can be calculated by subdividing the interval (0,H1) into the subintervals (0,  z) and (z,H1), and substituting Eqs. (34b) and (34a) for G in each case, respectively, whereas F2 can be calculated by substituting Eq. (36a) for G. Note the expressions for G do not depend on the spatial structure of the heating function Q, and can be used with arbitrary Q, provided it is diurnally oscillating. In section 4 we consider an alternative Q function emulating upper-level convective heating.

The solution for ψ2 proceeds almost identically to that for ψ1. As in the solutions of Rotunno (1983) and Qian et al. (2009), ψ is then recovered from ψ˜=ψ^1eit+ψ^2e−it by calculating the inverse Fourier transform in Eq. (20) numerically. As in previous studies we use a simple trapezoidal rule. The overall solution procedure may thus be summarized as follows.

  1. Nondimensionalize and combine the governing equations into a single partial differential equation (PDE) for the streamfunction ψ.

  2. Perform a Fourier transform with respect to x and consider the e±it modes separately.

  3. Solve the resulting PDE analytically using Green’s method.

  4. Recover ψ by performing the inverse Fourier transform numerically using the trapezoid rule.

We now consider an illustrative example solution, presented in dimensional coordinates to aid intuition. We state the governing parameters in dimensional coordinates, with the corresponding nondimensional numbers provided in Table 1. We set the coastal width parameter L = 50 km, the heating depth H = 1 km, Brunt–Väisälä frequencies N1 = 0.01 s−1 and N2 = 0.03 s−1, giving N=3,H1=2km, and the coastline occurring at the equator so that f = 0. Following Du et al. (2019), a representative heating amplitude of Q0 = 1.2 × 10−5 m s−3 was derived by integrating Eq. (6) at the surface and far inland between sunrise and sunset, assuming a temperature change of 10 K.

Table 1.

The parameters used for the various example solutions presented in this study, noting f is the Coriolis parameter and ω the angular frequency of Earth. See text for the definitions of each parameter and nondimensional number. Nonapplicable parameters are marked with N/A. The surface and convective heating functions are given by Eqs. (6) and (55), and depicted in Figs. 1 and 12, respectively.
Table 1.

Figure 2 depicts the example solution’s coastline perpendicular horizontal velocity u*. Close to the coastline x*=0 and near the surface the solution resembles that of Rotunno (1983), with a land breeze evident at 1200 LST (noon), and persisting until the sea breeze becomes established at the surface around 1500 LST, with the largest values of u* occurring around 2100 LST. At 0000 LST (midnight, not shown), Figs. 2a–d repeat, but with colors inverted. As in the solution of Rotunno (1983), wave rays emanate from the origin, and propagate both offshore and inland. However, unlike in the solution of Rotunno (1983), the rays are partially reflected and partially refracted at H1. Where the partially reflected ray meets the surface, it is perfectly reflected, producing echoes of the sea breeze far offshore and far inland from the sea breeze proper. Further still from the coastline the twice reflected ray again partially reflects and partially refracts at H1. This continues indefinitely, but with a little more energy escaping from D1 to D2 with each refraction, so that for all z ∈ D1, u* (x,z)→0 as x → ±∞. This behavior is similar to that for the low-level inversion, modeled by a discontinuity in potential temperature, considered by Jiang (2012b).

Fig. 2.
Fig. 2.

Coastline perpendicular vertical cross sections of the piecewise-constant solution for the coastline perpendicular horizontal winds in dimensional coordinates u* at, (a) 1200, (b) 1500, (c) 1800, and (d) 2100 LST, noting that these times correspond to values of 0, 3, 6, and 9 h for t*, respectively. The horizontal coordinate x* gives distance from the coastline. The coastal width is L = 50 km, the heating depth H = 1 km, and Brunt–Väisälä frequencies are N1 = 0.01 s−1 and N2 = 0.03 s−1, giving N=3. The change in stability occurs at z*=H1=2km, as indicated by the horizontal dotted line. The heating amplitude is Q0 = 1.2 × 10−5 m s−3.

Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-23-0074.1

Figures 3a and 3b decompose the u* winds at 2100 LST shown in Fig. 2d into those resulting from F1 and F2, respectively, i.e., from heating below and above H1, respectively. Given the structure of Q*, heating is concentrated in the first 1 km of the atmosphere, and hence, the overall solution is almost entirely determined by F1. The response to the upper-level forcing shown in Fig. 3b resembles a Saint Andrew’s cross, but with the rays below H1 reflecting off the lower boundary, and successively reflecting and refracting at H1 as before. In the supporting online material we consider an example where F2 is large enough to affect the overall solution.

Fig. 3.
Fig. 3.

As in Fig. 2, but for (a) the horizontal velocities u* resulting from heating F1 below the stability change at H1 = 2 km and (b) heating F2 above H1. Note the difference in color-bar scales, with the overall solution depicted in Fig. 2 determined almost entirely by F1.

Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-23-0074.1

Figures 4a and 4b illustrate the vertical velocity w* and the velocity field (u*, w*), respectively, at 1800 LST (sunset). In both cases the reflection and refraction behavior discussed above is clear. Away from the surface, and away from the change in stability z*=H1, velocity vectors along the wave rays are essentially parallel with the rays themselves, in accordance with theory. Near the surface, and for values of z just below H1, the velocity field represents the superpositioning of rays reflecting off the surface, and off the change in stability at z*=H1, respectively.

Fig. 4.
Fig. 4.

As in Fig. 2, but for (a) the vertical velocity in dimensional coordinates w* and (b) the velocity field in dimensional coordinates (u*, w*), with arrows and shading illustrating the direction and magnitude of the velocity vectors.

Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-23-0074.1

The reflection, refraction, and ducting coefficients defined above apply to the amplitudes of the streamfunctions and vertical velocities associated with individual waves. For the horizontal velocities, the refraction coefficient acquires an additional factor of N in the numerator, and the ducting coefficient in the denominator. Furthermore, while the ducting coefficient depends on m=κ/A, the reflection and refraction coefficients do not. The linearity of the inverse Fourier transform therefore implies that, far away from the forcing, the amplitudes of the reflected and refracted rays are also governed by r1 and r2, as evident in Fig. 2. Furthermore, the reflection, refraction, and ducting behaviors all follow from the Green’s function G, which is independent of the spatial structure of the heating function Q, implying these effects will generalize to other diurnal heating functions.

3. Transition-layer solution

In this section we consider how the solution presented in section 2 changes when the stability N varies linearly between N1 and N2 over a transition-layer H1≤z*<H2, rather than experiencing a discontinuity. Explicitly, we assume
N(z)={N1,0≤z*≤H1,N1+(z*−H1)M,H1<z*≤H2,N2,H2<z*,
where M=(N2−N1)/(H2−H1) is the slope of N in the transition layer. Equation (23) may then be derived as in the piecewise-constant case, but with the function n(z) now given by
n(z)={1,0≤z≤H1,1+(z−H1)M,H1<z≤H2,N,H2<z,
where H1=H1/H, H2=H2/H, N=N2/N1, and M=(N−1)/(H2−H1). We define the vertical subdomains D1=[0,H1],DTL=(H1,H2],D2=(H2,∞). Analogously to the piecewise-constant N case, we require that ψ^ and ψ^z be continuous at both H1 and H2.
Before solving Eq. (23), consider its unforced analog, Eq. (24). For z ∈ D1 and z ∈ D2, Eq. (24) reduces to the same pair of wave equations as the piecewise-constant N case, with analogous general solutions. However, for z ∈ DTL, Eq. (24) becomes
ψ^zz+m2[1+(z−H1)M]2ψ^=0.
This is a form of Weber’s equation, the general solution of which is the linear combination of two parabolic cylinder functions (e.g., Whittaker and Watson 1996) Da(z) and Db(z), where
Da(z)=D−(1/2)[(1+i)Z(z)],
Db(z)=D−(1/2)[(1−i)Z(z)],
with D−1/2 the parabolic cylinder function of argument −1/2, noting we have defined the convenience function Z:[H1,H2)→[m/|M|,mN/|M|) by
Z(z)=m|M|[1+M(z−H1)],
so that Z is both positive and real. To see that Da(z) and Db(z) satisfy Eq. (41), note that for an arbitrary variable ζ∈C, the function Dν(ζ) satisfies the differential equation (e.g., Wünsche 2003)
(d2dζ2−ζ24+ν+12)Dν(ζ)=0.
In the case of ν=−1/2, Eq. (45) reduces to
(d2dζ2−ζ24)D−1/2(ζ)=0.
Substituting the variable ζ = (±1 ±i)Z(z), it follows
d2dζ2D−(1/2)[(±1±i)Z(z)]=−m2[1+M(z−H1)]×D−(1/2)[(±1±i)Z(z)].
Thus, Da(z) and Db(z) satisfy Eq. (41). Physically, the functions Da(z) and Db(z) describe how upward- and downward-propagating waves are modulated as they pass through the transition layer: further details are provided in appendix A and the online supplement.

Equation (23) is again solved using Green’s method. The Green’s function G can be obtained by considering the z′ ∈ D1, z′ ∈ DTL, and z′ ∈ D2 cases separately, as in the piecewise-constant N case. The resulting expressions are analogous to those for the piecewise-constant solution, but more complex to state, and so are relegated to appendix B. Equation (32) can then be decomposed ψ^=F1+FTL+F2, analogously to before, where F1, FTL, and F2 now result from the heating below, within, and above the transition layer, respectively, noting FTL must be calculated numerically. We can then solve for ψ, u and w by calculating the inverse Fourier transforms numerically as before.

Figure 5 provides an illustrative example solution for the coastline perpendicular horizontal velocity u*, with the same choices of parameters as in the piecewise-constant example solution depicted in Fig. 2, but with a transition layer starting at z*=H1=1.5km and ending at z*=H2=2.5km (see Table 1). The behavior of the horizontal winds in Fig. 5 near the surface, close to the coastline, is essentially identical to the piecewise-constant solution. However, away from the surface, and away from the coastline, the behavior of the rays is notably different. The rays now refract smoothly, with the slope of the rays changing gradually over the transition layer. Moreover, ray reflection is substantially reduced, being only just visible in the weakest red and blue shaded regions. For transition-layer solutions corresponding to larger values of N (not shown), ray reflection becomes more evident, but the amplitude of the reflected ray is always less than in the corresponding piecewise-constant solution.

Fig. 5.
Fig. 5.

As in Fig. 2, but for the transition-layer solution for the coastline-perpendicular horizontal winds u*. The transition layer begins at z*=H1=1.5km and ends at z*=H2=2.5km, as depicted by the horizontal dotted lines, with N1 = 0.01 s−1 and N2 = 0.03 s−1, so that N=3, as before.

Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-23-0074.1

Figures 6a–c decompose the u* winds at 2100 LST shown in Fig. 5d into those resulting from F1, FTL, and F2, respectively, i.e., from heating below, within, and above the transition layer. As in the piecewise-constant solution, the overall response is almost entirely determined by F1, but ray reflection is substantially reduced, as noted above. The responses to FTL and F2 resemble the Saint Andrew’s cross, but with smooth ray refraction through the transition layer.

Fig. 6.
Fig. 6.

As in Fig. 5, but for (a) the horizontal velocities u* resulting from heating F1 below H1 = 1.5 km, (b) heating FTL within the transition layer between H1 = 1.5 km and H2 = 2.5 km, and (c) heating F2 above H2. Note the difference in color-bar scales, with the overall solution depicted in Fig. 5 determined almost entirely by F1.

Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-23-0074.1

Figures 7a and 7b illustrate the vertical velocity w* and the velocity field (u*, w*), respectively, at 1800 LST (sunset). The smooth refraction behavior is again evident, with less overall ray reflection than the piecewise-constant solution. Furthermore, the maximum speeds along the rays in D2 are ≈0.5 m s−1 faster in the transition-layer solution than in the piecewise-constant solution depicted in Fig. 4b, representing an increase of ≈15%.

Fig. 7.
Fig. 7.

As in Fig. 4, but for the transition-layer solutions for (a) w* and (b) the velocity field (u*, w*).

Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-23-0074.1

To determine the generality of the observed differences between the piecewise-constant and transition-layer solutions, consider again the reflection, refraction, and ducting coefficients, r1, r2, and r3, this time obtained by taking ratios of wave amplitudes in D1 and D2 from the expressions given in appendix B. Figure 8 provides example values for these coefficients for H1=2,N=3, and a range of values for H2≥H1, and the nondimensional vertical wavenumber m. As discussed in appendix B, these coefficients converge to their values in the piecewise-constant solution as H2→H1, but as H2−H1→∞ we also have the limits,
r1→0,r2→1N,r3→N.
We will refer to the H2→H1 and H2−H1→∞ limits as the short and long limits, respectively. The long limits are plotted as dashed gray lines in Figs. 8a–d, and it is evident that for larger wavenumbers, the transition-layer depth H2−H1 does not need to be particularly large before r1, r2, and r3 are approximately equal to their long limits. In appendix B we derive conditions for approximate convergence to the long limit: for N>1,
λH2−H1≪16π3(N−1),
and for N<1,
λH2−H1≪16πN23(1−N),
where λ=2π/m is the nondimensional vertical wavelength in D1. For example, for the m = 4 coefficients plotted in brown in Figs. 8a, 8b, and 8d, inequality (49) reduces to H2−H1≫3/16=0.19, and approximate convergence therefore occurs when H2−H1 is about an order of magnitude larger than 0.19, i.e., H2>H1+2=4. Figures 8a, 8b, and 8d indicate this is indeed the case, noting that r1 converges more slowly than r2 and r3.
Fig. 8.
Fig. 8.

Plots illustrating how (a) the reflection coefficient r1, (b) the refraction coefficient r2, and (c),(d) the ducting coefficient r3, change as the transition layer deepens, for various values of the nondimensional vertical wavenumber m, noting m=m*H, where m* is the wavenumber in dimensional coordinates, and H is the heating depth. The transition layer begins at z=H1=2, with N=3. Dashed gray lines show the large H2−H1 limit values 0, 1/N, and N for the coefficients r1, r2, and r3, respectively.

Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-23-0074.1

Similarly, approximate convergence to the short limit occurs when
λH2−H1≫πN2N−1
for N>1, and
λH2−H1≫π1−N
for N<1. Again considering the m = 4 coefficients in Fig. 8, inequality (51) reduces to H2−H1≪1/9, so that H2<2.01. Thus, for m = 4, the reflection, refraction, and ducting coefficients depart from their short limit values substantially, for even a small transition layer.

The short limit expression for the ducting coefficient r3 depends on both m and H1, and for particular choices of these values, the short limit for r3 may be either larger or smaller than the long limit. However, for the reflection and refraction coefficients r1 and r2, the long and short limit expressions depend only on N, not on m, H1, or H2. Furthermore, the short limit for the reflection coefficient r1=|(1−N)/(1+N)| is strictly greater than the long limit r1 = 0 for N>0,N≠1. Analogously, the short limit for the refraction coefficient r2=2/(N+1) is strictly less than the long limit r2=1/N for N>0,N≠1. While r1 and r2 oscillate as the depth of the transition layer increases, suggesting resonance between the vertical wavelength and the depth of the transition layer, r1 and r2 are bounded by their long and short limit values. Thus, waves reflect less, and refract more, in the presence of a transition layer, than when stability changes discontinuously.

To provide physical intuition for this result, consider again the problem of an unforced atmosphere with no lower boundary, a step change in stability at H1, and a radiation condition imposed above H1. Suppose first N>1, and that instead of both upward- and downward-propagating waves in D1, there is just a single wave in D1. At a height sufficiently far below H1, vertical parcel displacements associated with this wave are contained entirely in D1, but as z approaches H1, vertical parcel displacements impinge on D2. Because stability increases in D2, the total upward displacement experienced by the parcel will be smaller if it enters D2 than if stays entirely within D1. Specifically, a parcel oscillating about z=H1 is displaced a greater distance downward from H1 than upward. However, because the upward and downward phases of its oscillation must each occur over the same length of time, i.e., half the diurnal period, the vertical velocity w must be discontinuous at H1. But this violates conservation of mass: the continuity equation ux + wz = 0 cannot be satisfied as wz would then be infinite at H1, whereas ux would remain finite. Thus, to conserve mass there must be a second wave in D1, i.e., the reflected wave, which offsets the negative vertical displacements associated with the incident wave.

Consider now the analogous transition-layer case, where stability increases by the same amount overall, but the change occurs linearly between H1 and H2, again assuming N>1. Suppose there is just a single wave in D1. As z approaches H1, parcels displaced upward from z must move through the transition layer DTL before potentially reaching D2, with the stability throughout DTL lower than that in D2. For a parcel originally at H1, the upward displacement is again smaller in magnitude than the downward displacement, but larger in magnitude than the upward displacement when stability is piecewise constant. Thus, to conserve mass, the amplitude of the reflected wave must be smaller in the transition-layer case than in the piecewise-constant case, as the difference between the magnitudes of the upward and downward displacements is smaller in the latter case than the former. This argument also shows why the amplitude of the reflected wave must approach zero as H2−H1→∞. Analogous arguments apply when N<1. Linear displacement fields for the piecewise-constant and transition-layer example solutions are provided in the online supplement, which assist in visualizing the preceding arguments.

While the limits in Eq. (48) can be derived in a direct way mathematically, they can also be inferred indirectly, albeit informally. Consider individual sinusoidal wave perturbations (u, υ, w), i.e., perturbations whose spatial structure in x is given by eiκx for some wavenumber κ. Suppose these perturbations occur against a horizontal background wind (u¯,0,0). Under the Boussinesq approximation on an f plane, the temporal rate of change of the mean background-state horizontal velocity u¯ is given by the convergence of mean specific wave momentum flux (e.g., Sutherland 2010),
∂u¯∂t=−∂⟨uw⟩∂z,
where angle brackets denote the average over a horizontal wavelength in the x direction. Because u¯=∂u¯/∂t=0 by construction in our study, the mean specific momentum flux 〈uw〉 must be constant in z. Suppose the amplitude of the vertical velocity of the incident wave is A. From the parcel argument given above, in the long limit the amplitude of the reflected wave is zero, so
⟨uw⟩={−A22,z∈D1,(54a)−NA2r222,z∈D2.(54b)
Thus, in the long limit, the refraction coefficient r2 must be 1/N to ensure the specific momentum flux is constant. Analogous considerations apply for the ducting coefficient r3.

For a transition layer of finite, nonzero thickness, r1 and r2 depend on m, and the reflection and refraction coefficients of the rays will lie somewhere between the long and short limit values for the component waves comprising the rays. As the thickness of the transition layer increases, inequalities (49) and (50) imply a greater proportion of the ray’s wavenumber spectrum behaves according to the long limit, rather than the short limit. Thus, the reflection and refraction coefficients for the rays will also approach the long limits 0 and 1/N as H2−H1→∞. Similarly, inequalities (51) and (52) imply the reflection and refraction coefficients of the rays converge to the short limit expressions for r1 and r2 as H2→H1. Analogous arguments apply to the ducting coefficient for the rays, with the amplitude of the ducted ray approaching N as H2−H1→∞. Analogously to the piecewise-constant solution, the reflection, refraction, and ducting behaviors in the transition-layer solution follow from the Green’s function G given in appendix B, implying these effects will again generalize to other diurnal heating functions Q.

4. Years of the Maritime Continent soundings

The original motive for formulating the transition-layer problem was to interpret a set of radiosonde observations obtained from the second Australian leg of the YMC field campaign during November and December 2019 (Yoneyama and Zhang 2020; Protat and McRobert 2020b; Protat et al. 2022). During this campaign, the R/V Investigator, Australia’s scientific research vessel, sailed through the waters near Darwin, the capital of Australia’s Northern Territory, as depicted in Fig. 9. Between 20 November and 2 December 2019, the Investigator traveled from the waters northwest of Darwin, past Croker Island, and back again, launching eight radiosondes per day as it did so.

Fig. 9.
Fig. 9.

Maps depicting, (a) Australia’s Northern Territory, its capital Darwin, and surrounding landmarks, and (b) the region within the red dashed box in (a), showing the location of the R/V Investigator between the 20 Nov and the 2 Dec 2019.

Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-23-0074.1

Figure 10 illustrates the vertical profile of the Brunt–Väisälä frequency N obtained from the soundings taken between 20 November and 2 December 2019. The light gray line depicts the profile of N at 0000 LST 20 November 2019, and the dark gray line the average over this day. The thicker black line provides the average over the entire 20 November–2 December 2019 time period, with a 200 m running mean applied to reduce the small-scale noise. Note the average profile of N is ≈0.01 s−1 below ≈15 km and ≈0.025 s−1 above ≈19 km, transitioning roughly linearly between these two values between these two heights. This structure is not an artifact of the averaging: it is also present, albeit in a noisier form, in the gray lines. Furthermore, an N transition layer is consistent with previous work demonstrating that under various tropopause definitions utilizing diverse variables, the tropopause resembles a transition layer, rather than a discontinuity (Pan et al. 2004; Schmidt et al. 2006; Fueglistaler et al. 2009; Feng et al. 2012).

Fig. 10.
Fig. 10.

The vertical profile of the Brunt–Väisälä frequency N from the soundings obtained during the portion of the R/V Investigator voyage depicted in Fig. 9b. The light gray line depicts the Brunt–Väisälä frequency at 0000 LST 20 Nov 2019, the dark gray line depicts the average over 20 Nov 2019, while the thicker black line depicts the average over the entire 20 Nov–2 Dec 2019 period, with a 200 m running mean applied to this average to further smooth small-scale noise.

Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-23-0074.1

Figure 11a provides a Hovmöller diagram of the meridional winds from the soundings between 0000 LST 20 November and 0000 LST 2 December 2019, with winds linearly interpolated in time onto a regular hourly time step. White regions indicate missing values, where the radiosonde’s balloon burst before ascending above 25 km. Note there is significant vertical shear in the meridional winds, particularly between ≈8 and 17 km. Recall that we assumed background winds were zero in the theory presented in sections 2 and 3, as sheared background winds significantly complicate the analytic approach (Du et al. 2019): our theory should therefore be applied cautiously to this YMC dataset. Nevertheless, note the weak stratospheric meridional background winds above ≈17 km in Fig. 11a, with clear downward-propagating diurnal perturbations.

Fig. 11.
Fig. 11.

Hovmöller diagrams of (a) the meridional winds, (b) the meridional wind perturbations against a 24 h centered-running-mean background wind, and (c) the average perturbations at each time of day, from the soundings taken on board the R/V Investigator between 20 Nov and 2 Dec 2019. The location of the R/V Investigator during this period is depicted in Fig. 9b.

Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-23-0074.1

Figure 11b provides the corresponding Hovmöller diagram of the wind perturbations against a 24-h centered running mean background wind, with missing values interpolated linearly when calculating the background wind, but not the perturbations. Diurnal perturbations are now evident throughout both the stratosphere and troposphere. Sometimes the tropospheric perturbations also propagate downward, but at a faster vertical phase speed than those in the stratosphere, while at other times they are essentially stationary. Figure 11c provides a Hovmöller diagram of the composite diurnal cycle, obtained by averaging the perturbations depicted in Fig. 11b at each hour of the day: the cycle is repeated once for clarity. A downward-propagating signal is again evident in the stratosphere, with the signal more stationary below 15 km, and showing a degree of discontinuity at ≈4 km. (Note that in Figs. 17b and 17c, the magnitude of the winds above 17 km are of comparable or greater magnitude than those below 17 km.)

Suppose we interpret the perturbations in Figs. 11b and 11c as representing gravity wave rays forced at the diurnal frequency. Such waves would likely have multiple sources, including the immediate Australian coastline, remote coastlines, and nearby and remote diurnal cycles of convection. The basic solution of Rotunno (1983), and Figs. 2 and 5, suggest that wave rays forced at the surface along a coastline would not reach the stratosphere until thousands of kilometers away from that coastline. Over such distances, the rays would likely disperse through the effect of background winds, friction, and violations of the f-plane assumption.

Suppose then that the waves are forced nearby, but at upper levels. Indeed, Hankinson et al. (2014) applied ray-tracing methods to stratospheric gravity waves observed in radiosonde observations over Darwin, concluding the waves originated from convection over Indonesia, the Philippines, and New Guinea. In particular, Hankinson et al. (2014) attributed the lower-stratospheric waves with vertical wavelengths of 2–4 km, similar magnitudes to those of the stratospheric signals in Fig. 11, to New Guinea convection. Furthermore, in a high-resolution simulation, Vincent and Lane (2016, Fig. 15) documented wave rays forced along the northern New Guinea coastline, with additional rays forced at upper levels around 12 km altitude, roughly in phase with those forced near the surface. These upper-level rays were likely forced by the convective and stratiform heating associated with the convective line that occurs daily over the New Guinea mountain range during austral summer.

To emulate the upper-level heating associated with a convective line, we redefine
Q*=Q0exp⁡[−x*2L2−(z*−H)2D2]cos⁡(ωt*),
where L and D give the horizontal and vertical scales of a two-dimensional Gaussian heating function centered at x*=0, and z*=H, similar to the function considered by Robinson et al. (2008). Figure 12 provides an example of the new heating function, with L = 100 km, H = 12 km, D = 4 km, and Q0 = 6 × 10−6 m s−3.
Fig. 12.
Fig. 12.

The heating function given by Eq. (55) at 1200 LST, i.e., t*=0, with L = 100 km, H = 12 km, D = 4 km, and Q0 = 6 × 10−6 m s−3.

Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-23-0074.1

With the heating function given by Eq. (55), Eq. (32) becomes
ψ^(z)=12A2iπkLe−(k2L2/4)∫0∞G(z,z′)e−[(z′−1)2/D2]dz′,
where L=ωL/(N1H) and D=D/H, with the Green’s function G given by the same expressions as in section 2 and appendix B. Analytic solutions to Eq. (56) can then be given in terms of the Gaussian error function, with the exception of the transition-layer solution when z′ ∈ DTL, which must be solved numerically as before.

Figure 13 presents an example piecewise-constant solution using the revised heating function, with H1 = 17 km, L = 100 km, D = 4 km, and H = 12 km (see Table 1). It is somewhat unclear how Q0 should be chosen: as a first guess we take Q0 = 6 × 10−6 m s−3, half that of the surface forcing considered in sections 2 and 3, with this choice motivated by the potential temperature tendencies and perturbations in the simulation and observational results of Vincent and Lane (2016, 2018). From Fig. 10 we take N1 = 0.01 s−1 and N2 = 0.025 s−1, so that N=2.5. We choose an f plane based on a constant latitude of 11.5°S.

Fig. 13.
Fig. 13.

The piecewise-constant solution for the convective-line-perpendicular horizontal winds u*, using the heating function given by Eq. (55), with H1 = 17 km, L = 100 km, D = 4 km, H = 12 km, and Q0 = 6 × 10−6 m s−3. The Brunt–Väisälä frequencies are N1 = 0.01 s−1 and N2 = 0.025 s−1, so that N=2.5, and the f plane is chosen based on a constant latitude of 11.5°S. Depicted are the winds at, (a) 1200, (b) 1500, (c) 1800, and (d) 2100 LST. The vertical gray dashed lines at x*=−600,−800, and−1000km show the vertical transects considered in Fig. 17.

Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-23-0074.1

A Saint Andrew’s cross pattern is evident in Fig. 13, with reflection and refraction of the upper rays at H1. The reflected rays, which are narrow, superpose on the lower rays forced at 12 km. Vertical phase speeds associated with the upper rays are negative as before, but positive for the lower rays, as required by the radiation condition. Also, the horizontal winds are now out of phase either side of x*=0, physically consistent with the convergence/divergence associated with a localized heating function: mathematically, this occurs because Qx is now a dipole rather than a monopole.

Figure 14 decomposes the 1500 LST horizontal winds depicted in Fig. 13b into those arising from F1 and F2, respectively, i.e., forcing below and above the stability change at H1 = 17 km. As in section 2, the overall response is mostly determined by F1, recalling that the heating is concentrated at H = 12 km.

Fig. 14.
Fig. 14.

As in Fig. 13, but for (a) the horizontal velocities u* at 1500 LST resulting from heating F1 below the stability change at H1 = 17 km, as depicted by the horizontal dotted line, and (b) heating F2 above H1. Note the difference in color-bar scales, with the overall solution depicted in Fig. 13 determined mostly by F1.

Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-23-0074.1

Figure 15 provides the corresponding transition-layer solution, with H1 = 15 km and H2 = 19 km (see Table 1). As before, the rays reflect less than in the piecewise-constant solution, with more energy escaping into the stratosphere. Figure 16 decomposes the 1500 LST u* winds depicted in Fig. 15b into those arising from F1, FTL, and F2, respectively, i.e., heating below, within, and above the transition layer between H1 = 15 km and H2 = 19 km. Although the reflected ray in Fig. 16 is seemingly of larger magnitude than the reflected ray in the piecewise-constant solution in Fig. 14a, the domains D1 encompass different heights in each case, and when considered with the response to the heating within the transition layer FTL, the apparent increased reflection is almost completely offset, so that the net reflection is less than in the piecewise-constant case.

Fig. 15.
Fig. 15.

As in Fig. 13, but for the transition-layer solution, with H1 = 15 km and H2 = 19 km, as depicted by the horizontal dotted lines.

Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-23-0074.1

Fig. 16.
Fig. 16.

As in Fig. 15, but for the horizontal velocities u* at 1500 LST resulting from (a) heating F1 below H1 = 15 km, (b) heating FTL within the transition layer between H1 = 15 km and H2 = 19 km, and (c) heating F2 above H2. Note the difference in color-bar scales, with the overall solution depicted in Fig. 15 determined mostly by F1 and FTL.

Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-23-0074.1

To qualitatively compare the solutions above with the YMC soundings, suppose we interpret x*=0 as the position of a daily recurring convective line oriented roughly perpendicular to the displacement vector from the convective line to the R/V Investigator. Following the discussion above, we might take x*=0 as the position of the New Guinea mountain range, with the y* direction then extending along the range. We could then view the R/V Investigator as being located some distance in the negative x* direction, and the meridional winds depicted in Fig. 11 as loosely corresponding to the convective-line-perpendicular winds u* in Figs. 13 and 15.

Figure 17a provides a Hovmöller diagram of the u* winds along the vertical transects at x = −600, −800, and −1000 km, depicted by the dashed gray lines in Figs. 13 and 15. There are qualitative similarities, but also notable differences, between the Hovmöller diagrams in Figs. 11 and 17. Each of the diagrams in Fig. 17 features negative phase speeds above 12 km, consistent with Fig. 11. In the transition-layer solution, the horizontal wind speeds above 17 km are at least 0.25 m s−1, or ≈20%, larger than in the piecewise-constant solution. Moreover, the winds directly above and below 17 km are of comparable magnitude, and the overall signal smoother, compared with the piecewise-constant solution, which is more erratic due to increased reflection and discontinuous refraction. This suggests the transition-layer solution more realistically simulates the transfer of energy through the tropopause than the piecewise-constant solution.

Fig. 17.
Fig. 17.

Hovmöller diagrams of the convective-line-perpendicular horizontal winds u* from (a),(c),(e) the piecewise-constant solution and (b),(d),(e) the transition-layer solutions depicted in Figs. 13 and 15 along the gray dashed vertical transects at (a),(b) x*=−600km, (c),(d) x*=−800km, and (e),(f) x*=−1000 km. The heights of the stability changes H1 and H2 are depicted by the horizontal dotted lines.

Citation: Journal of the Atmospheric Sciences 80, 10; 10.1175/JAS-D-23-0074.1

Timings and magnitudes of the upper-level signals are comparable to those in Fig. 11, although the upper-level response ends below 20 km in Fig. 17, but extends to at least 25 km in Fig. 11. One explanation for this difference is the lack of background winds in the theory, with background winds in the convective-line-perpendicular direction acting to partially disperse the rays, resulting in more oscillations in the vertical (Qian et al. 2009).

A major difference between Figs. 11 and 17 is in the response below 12 km. In Fig. 17 the perturbations below 12 km exhibit positive vertical phase speeds as discussed above, whereas in Fig. 11 phase speeds are negative, or the signal is stationary. One explanation for this difference is that, in reality, additional surface or low-level convective diurnal heating is present, generating rays with negative phase speeds which superpose with the downward pointing rays with positive phase speeds generated by the upper-level forcing. This superpositioning could result in a signal that is stationary, or downward propagating. The apparent continuity of the upper and lower perturbations in Figs. 11b and 11c would then be essentially coincidental. These ideas could be critically tested using ray-tracing methods applied to more complex numerical model data (e.g., Hecht et al. 2004; Alexander et al. 2004; Vincent et al. 2004; Hankinson et al. 2014).

To generalize the above ideas, note that the line-parallel winds υ* can be recovered from the line-perpendicular winds u* using polarization relations inferred from Eq. (11), which show that υ* is in quadrature with u*, with the amplitude of υ* equal to that of u* scaled by f/ω. The horizontal winds in other directions then comprise weighted sums of u* and υ*, and hence, the essential structure of the solutions depicted in Figs. 13 and 15 will not change if other wind directions are considered. Analogous points apply to rotations of the convective line in the x*–y* plane, excluding the degenerate cases of rotations by ±90°.

5. Discussion and conclusions

In this study we extended the tropical linear sea-breeze theory of Rotunno (1983) to situations involving nonconstant Brunt–Väisälä frequency N. We first presented an illustrative example of a low-level stability change, emblematic of that between the boundary layer and troposphere. Because solutions are formulated using the Green’s function G, generalizing to other heating functions Q is straightforward. We therefore also considered an alternative heating function representing diurnally recurrent convective or stratiform heating aloft, and considered how forced waves behave as they passed through the tropopause.

In both cases wave reflection, refraction, and ducting behavior depends not only on the height and magnitude of the stability change, but on the thickness of the transition layer, at least in the idealized circumstances of the theory. This behavior follows from the structure of the Green’s function G, implying broad generality of this result. It may therefore be instructive to consider how the solutions presented here behave across diverse climatic conditions, and to compare this behavior with more realistic numerical simulations, or observations. For example, increases in tropopause height are anticipated under warming (e.g., Hu and Vallis 2019), and this will affect the tropopause reflection, refraction, and ducting behavior in the theory presented here. Furthermore, the thickness of the tropopause transition layer varies with latitude and season, and there is an observed thickening trend over the last four decades (Schmidt et al. 2006; Feng et al. 2012), with idealized models indicating the structure of the tropopause transition layer is sensitive to tropospheric warming, stratospheric cooling, and the vertical ozone profile (Lin et al. 2017; Dacie et al. 2019). The solutions presented in this study therefore suggest tropopause wave reflection, refraction, and ducting will vary significantly across season and latitude, and potentially also with climate. These hypotheses could be investigated with more realistic modeling experiments.

The solutions also provide test cases for assessing the numerical methods used in more complex models. In particular, if the vertical discretization of a finite-difference-based time stepping numerical method is too coarse over a stability change, we might expect simulated wave rays to reflect and refract as in the piecewise-constant solution, rather than the appropriate transition-layer solution, with finite-difference-based schemes therefore exaggerating the total energy reflected. In the online supplement we provide a preliminary investigation of these ideas.

Simplifying assumptions were made to derive the linear equations considered in this study, as in the related studies described in section 1. Most significant is perhaps linearity itself, which eliminates density current dynamics, an important aspect of the land–sea breeze at low levels near the coast (Qian et al. 2012). Nonlinear and nonhydrostatic effects may also be significant in certain contexts (e.g., Pandya et al. 1993, 2000). Furthermore, viscosity may increasingly attenuate wave rays with horizontal distance from their source (Yan and Anthes 1987; Dalu and Pielke 1989), so that if the stability change occurs a large vertical distance from the forcing, reflection and refraction become a moot point.

Another potential limitation is the Boussinesq approximation, which is usually thought to be valid only over vertical scales much less than the density scale height Hρ, where Hρ ≈ 8 km in the troposphere. However, Du et al. (2019) compared a similar set of analytic solutions, involving a nonconstant background wind u¯(z) but constant N, to two-dimensional Weather Research and Forecasting (WRF) Model simulations, and found a remarkable agreement between the two throughout the lowest 10 km of the atmosphere. This agreement, in conjunction with the scale analysis of Ogura and Phillips (1962), suggest that in the atmosphere it is actually the potential temperature scale height Hθ ≈ 30 km that determines the validity of the Boussinesq approximation. We therefore expect our core results about reflection, refraction, and ducting will be unaffected by the Boussinesq approximation, provided the depth of the transition layer is small compared to Hθ: this could be tested using a similar set of WRF experiments as Du et al. (2019).

The most physically important process missing from the theory presented here is probably background winds, as these not only alter the basic structure of the wave rays (Qian et al. 2009), but changes in background winds with height also induce attenuation, reflection, and refraction (Du et al. 2019; Lane 2021) of their own. A straightforward extension of our piecewise-constant solution would be to ignore the Coriolis term, but include constant background winds above and below the stability change, with a discontinuity permitted at the stability change. Analogously, the transition-layer solution could be extended by ignoring Coriolis, but allowing the background winds to shear linearly between the constant values above and below the transition layer. Equation (41) would then become a form of Whittaker’s equation, with Whittaker function solutions, which may permit a similar analysis to that of the present paper.

To summarize, in this study we began by extending the linear sea-breeze theory developed by Rotunno (1983) to include vertical changes in the Brunt–Väisälä frequency N. In section 2 we presented the solution in the case where N = N1 below some height H1, and N = N2 above H1, where N1 and N2 are constant. The behavior of this solution is determined by the nondimensional parameters A,L,N=N2/N1, and H1=H1/H. In section 3 we presented the solution for the case where N = N1 below some height H1, N = N2 above some other height H2 > H1, with N transitioning linearly between these two values for z* between H1 and H2. The behavior of this solution is determined by the nondimensional parameter H2=H2/H, in addition to those for the piecewise-constant solution.

In the piecewise-constant solution, gravity wave rays emanate from the origin, as in the base solution of Rotunno (1983). These rays are generated by heating below H1. The individual waves comprising the rays reflect and refract, with the amplitudes of the reflected and refracted waves governed by the coefficients r1=|(1−N)/(1+N)| and r2=2/(N+1), respectively. These expressions are the Boussinesq versions of the coefficients derived by Lindzen and Tung (1976). Because these coefficients are independent of the nondimensional vertical wavenumber m, the amplitudes of the reflected and refracted rays are also governed by r1 and r2.

When stability decreases with height, or H1 is comparable or smaller than the vertical scale of the heating H, waves forced by heating above H1 also play a significant role. Below the forcing, the rigid lower boundary implies vertically nonpropagating waves are present above and below H1, with the ratio of their amplitudes given by the ducting coefficient r3=N/cos⁡2(mH1)+N2sin⁡2(mH1), or equivalently, r3=N/cos⁡2(m*H1)+N2sin⁡2(m*H1), where m*=m/H is the vertical wavenumber in dimensional coordinates. Because r3 depends on m, the ducting behavior of the ray depends on the specifics of its wavenumber spectrum.

Overall, the reflection, refraction, and ducting behavior in the transition-layer solution is significantly different from the piecewise-constant solution. The coefficients r1, r2, and r3 now depend on m, but for each m the amplitudes of the reflected wave is lower, and the amplitude of the refracted wave greater, than in the corresponding piecewise-constant solution. The reflection, refraction, and ducting coefficients approach their piecewise-constant expressions when H2 → H1. Furthermore, the reflection, refraction, and ducting coefficients approach the limits 0, 1/N, and N as H2 − H1 → ∞. Because these long limits do not depend on m, as H2 − H1 → ∞ the limiting amplitudes of the reflected, refracted, and ducted rays are also governed by these coefficients. Away from the forcing, the amplitude of the reflected ray is therefore always lower, and refracted ray higher, in the presence of a transition layer, than a discontinuity in stability. The change in stability N does not need to be particularly extreme, nor the thickness of the transition layer large, before the behavior of the rays moves away from the piecewise-constant limit, and substantially toward the large limit.

The core reflection, refraction, and ducting behaviors described above all follow from the structure of the Green’s function G, not the spatial structure of the heating function Q, and it is therefore straightforward to consider other functions Q. In section 4 we considered an alternative heating function which emulates the upper-level convective or stratiform heating associated with a convective line. In this case H is the height at which heating is concentrated, and the solution depends on an additional nondimensional parameter D=D/H, which describes the vertical depth of the heating. We compared this modified theory to observations taken during the Australian leg of the YMC field campaign, finding the observations showed better qualitative agreement with the transition-layer solution than the piecewise-constant solution. For realistic values of N, and a realistic thickness of the tropopause transition layer, there is notably less reflection, and more refraction, in the transition-layer solution than the piecewise-constant solution. As noted by Lindzen and Tung (1976), when modeled as a step change in stability, the tropopause is “a rather poor reflector”: when modeled by a transition layer, it becomes an outright bad reflector.

Acknowledgments.

Funding for this study was provided for Ewan Short by the Australian Research Council’s Centre of Excellence for Climate Extremes (CE170100023), and by the Australian Bureau of Meteorology. Thanks are due to Alain Protat, chief scientist during the Australian leg of the YMC campaign, the Australian Marine National Facility, the R/V Investigator crew, and the YMC science team for the soundings used in this study.

Data availability statement.

The code written to generate and plot the solutions, and animated versions of key figures, are freely available online (Short 2023). The YMC sounding data are also freely available online (Protat and McRobert 2020a).

APPENDIX A

Parabolic Cylinder Functions

In this appendix we describe the functions Da(z) and Db(z). First note that parabolic cylinder functions can be expressed as the product of a “wavelike” exponential function, and a power series (e.g., Wünsche 2003), so that the solution to Eq. (41) can be written
ψ^=b1Da(z)+b2Db(z)
=b1e−i[Z(z)2/2]U−1/2[(1+i)Z(z)]+b2ei[Z(z)2/2]U−1/2[(1−i)Z(z)],
for constants b1,b2∈C, where
U−1/2[(1±i)Z(z)]=∑k=0∞(−1)kΓ(k+12)Γ(2k+34)k!e±ik(π/4)Z(z)k.
Alternatively (Wünsche 2003),
U−1/2[(1±i)Z(z)]=πΓ(34)1F1[14;12;±iZ(z)2]−2πΓ(14)(1±i)Z(z)1F1[34;32;±iZ(z)2]=πZ(z)2{e±i[Z(z)2/2]J−1/4[Z(z)22]−e±i[Z(z)2/2+π/4]J1/4[Z(z)22]},
where 1F1 is the confluent hypergeometric function of the first kind, which may be expressed using Bessel functions of the first kind J±1/4 (Abramowitz and Stegun 1972), with additional simplifications as Z(z) is real and nonnegative. Thus,
Re{U−1/2[(1±i)Z]}=πZ2[cos⁡(Z22)J−1/4(Z22)−J1/4(Z22)cos⁡(π4+Z22)],
Im{U−1/2[(1±i)Z]}=±πZ2[sin⁡(Z22)J−1/4(Z22)−J1/4(Z22)sin⁡(π4+Z22)],
so that
Re{U−1/2[(1+i)Z(z)]}=Re{U−1/2[(1−i)Z(z)]},
Im{U−1/2[(1+i)Z(z)]}=−Im{U−1/2[(1−i)Z(z)]},
and hence,
ψ^=b1Da(z)+b2Db(z)
=b1A(z)e−i[Z(z)2/2]+iθ(z)+b2A(z)ei[Z(z)2/2]−iθ(z),
where A(z) and θ(z) are real valued functions, corresponding to the amplitude and argument of the complex valued function U−1/2[(1±i)Z(z)], respectively. As a function of Z, θ(Z = 0) = 0, which follows from Eq. (A3), and remarkably, lim⁡Z→∞θ(Z)=−π/8, which follows from substituting the large asymptotic expressions for the Bessel functions in Eqs. (A5) and (A6) (e.g., Whittaker and Watson 1996).

Plots of A(z) and θ(z) are provided in the online supplement, and while difficult to prove formally, it is apparent that these functions do not oscillate for z ∈ DTL. Furthermore, in this form the functions Da(z) and Db(z) bear a close resemblance to the first-order Wentzel, Kramers, and Brillouin (WKB) approximate solutions to Eq. (41), and thus, the functions Da(z)eit and Db(z)e−it can be interpreted as modulated upward- and downward-propagating waves, respectively; further details are provided in the online supplement. In the form of Eq. (A10), algebra involving Da(z) and Db(z) is considerably simpler, and in their Bessel function forms, A and θ can be rapidly calculated to machine precision, simplifying numerical implementation.

APPENDIX B

Transition-Layer Expressions

As discussed in section 3, the transition-layer solution can be derived by separately solving for G(z, z′) for the subcases z′ ∈ D1, z′ ∈ DTL, and z′ ∈ D2. Here we present the expressions for G associated with ψ^1, the leftward-propagating mode: the expressions associated with the rightward-propagating mode ψ^2 are analogous.

In the derivation for the z′ ∈ D1 case, which is included in the online supplement, the following expressions arise:
α(z)=iZ(z)dZ(z)dz−idθdz
={im[1+M(z−H1)]−idθdz,N>1,−im[1+M(z−H1)]−idθdz,N<1,
β=imN+α(H2)−A′(H2)A(H2)2α(H2),
X=12[(1−β)Da(H1)Da(H2)+βDb(H1)Db(H2)],
Y=12im{(1−β)Da(H1)Da(H2)[−α(H1)+A′(H1)A(H1)]+βDb(H1)Db(H2)[α(H1)+A′(H1)A(H1)]},
P=(X+Y)eimH1+(X−Y)e−imH1,
where θ and A are defined in appendix A, and A′ denotes dA/dz. For z′ ∈ D1, G is then given by
G={−1mP[(X−Y)eim(z′−H1)+(X+Y)e−im(z′−H1)]sin⁡(mz),z∈D1,z≤z′,(B7a)−1mPsin⁡(mz′)[(X−Y)eim(z−H1)+(X+Y)e−im(z−H1)],z∈D1,z′<z,(B7b)−1mP[(1−β)Da(H2)Da(z)+βDb(H2)Db(z)]sin⁡(mz′),z∈DTL,(B7c)−1mPsin⁡(mz′)eimN(z−H2),z∈D2. (B7d)
These expressions are analogous to Eqs. (34a)–(34c). The amplitudes X − Y and X + Y appear in place of (N+1)/2 and (1−N)/2 in Eqs. (25) and (34a)–(34c), and approach these as H2→H1. The reflection and refraction coefficients are given by r1=|(X+Y)/(X−Y)| and r2=|1/(X−Y)|, respectively.
When solving for G in the z′ ∈ DTL case, the following expressions arise:
γ=mcos⁡(mH1)+sin⁡(mH1)[−A′(H1)A(H1)+α(H1)]2α(H1),
μ=γ(1−β)Db(H2)Da(H1)−[sin⁡(mH1)−γ]βDa(H2)Db(H1),
U(z′)=−Da(H2)Db(H1)Da(H1)β2μα(z′)Da(z′)−Db(H2)Db(H1)Da(H1)(1−β)2μα(z′)Db(z′),
V(z′)=−Da(H1)Db(H2)Da(H2)γ2μα(z′)Da(z′)−Db(H1)Db(H2)Da(H2)[sin⁡(mH1)−γ]2μα(z′)Db(z′).
For z′ ∈ DTL, G is given by
G={U(z′)sin⁡(mz),z∈D1,(B12a)U(z′)[[sin⁡(mH1)−γ]Da(H1)Da(z)+γDb(H1)Db(z)],z∈DTL,z≤z′,(B12b)V(z′)[(1−β)Da(H2)Da(z)+βDb(H2)Db(z)],z∈DTL,z′<z,(B12c)V(z′)eimN(z−H2),z∈D2.(B12d)
These expressions describe the vertical response to a point forcing within the transition layer DTL, and have no direct analog in the piecewise-constant solution. A vertically nonpropagating wave and a wave satisfying the radiation condition are evident in D1 and D2, respectively.
Finally, when solving for G when z′ ∈ D2, the following expressions arise:
T=12{[sin⁡(mH1)−γ]Da(H2)Da(H1)+γDb(H2)Db(H1)},
S=12imN{[sin⁡(mH1)−γ]Da(H2)Da(H1)[−α(H2)+A′(H2)A(H2)]+γDb(H2)Db(H1)[α(H2)+A′(H2)A(H2)]}.
For z′ ∈ D2, G is then given by
G={−12imN(S−T)eimN(z′−H2 )sin⁡(mz),z∈D1,     (B15a)−12imN(S−T)eimN(z′−H2){[sin⁡(mH1)−γ]Da(H1)Da(z)+γDb(H1)Db(z)},z∈DTL,           (B15b)−12imN(S−T)eimN(z′−H2)[(S+T)eimN(z−H2)−(S−T)e−imN(z−H2)],z∈D2,z≤z′,       (B15c)−12imN(S−T)[(S+T)eimN(z′−H2)−(S−T)e−imN(z′−H2)]eimN(z−H2),z∈D2,z′<z.    (B15d)
These expressions are analogous to Eqs. (36a)–(36c). The amplitudes S + T and S−T=S+T¯ appear in place of R and R¯, and approach these as H2→H1. The ducting coefficient is given by r3=|1/[2i(S+T)]|.

It is straightforward to show each of the above expressions for G approach their piecewise-constant forms as H2→H1, and those of the base solution of Rotunno (1983) as N→1. Also, as H2−H1→∞, the functions dθ/dz and dA/dz approach zero, and from the large asymptotic expressions for the Bessel functions in Eqs. (A5) and (A6), we can show |A(H2)/A(H1)| approaches N, from which the long limit expressions for the reflection, refraction, and ducting coefficients follow.

Note that Bessel functions J±1/4[Z(z)2/2] approximate their large asymptotic forms when Z2/2≫3/16 (Abramowitz and Stegun 1972), which requires min⁡[m/|M|,mN2/|M|]≫3/8, from which follows inequalities (49) and (50). Analogously, approximate convergence to the short limit occurs when max⁡[m/|M|,mN2/|M|]≪1, from which follows inequalities (51) and (52).

One final limiting behavior remains to be checked. Suppose that instead of n(z) varying linearly over the transition layer between 1 and N, the transition layer consists of d ≥ 2 discontinuities in N, with a constant difference between subsequent values of n(z), with the discontinuities equally spaced between H1 and H2. We would expect the solution in this case to approach that of the continuous linear transition layer as d → ∞. While difficult to prove in complete generality, in the online supplement we illustrate this behavior with examples. The wave modulation associated with Da(z) and Db(z) can thus be understood as the combined effect of many small reflections and refractions in the limit as the number of stability discontinuities becomes large.

REFERENCES

  • Abramowitz, M., and I. A. Stegun, 1972: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, 1076 pp.

  • Alexander, M. J., P. T. May, and J. H. Beres, 2004: Gravity waves generated by convection in the Darwin area during the Darwin Area Wave Experiment. J. Geophys. Res., 109, D20S04, https://doi.org/10.1029/2004JD004729.

    • Search Google Scholar
    • Export Citation

  • Bergemann, M., C. Jakob, and T. P. Lane, 2015: Global detection and analysis of coastline-associated rainfall using an objective pattern recognition technique. J. Climate, 28, 7225–7236, https://doi.org/10.1175/JCLI-D-15-0098.1.

    • Search Google Scholar
    • Export Citation

  • Dacie, S., and Coauthors, 2019: A 1D RCE study of factors affecting the tropical tropopause layer and surface climate. J. Climate, 32, 6769–6782, https://doi.org/10.1175/JCLI-D-18-0778.1.

    • Search Google Scholar
    • Export Citation

  • Dalu, G. A., and R. A. Pielke, 1989: An analytical study of the sea breeze. J. Atmos. Sci., 46, 1815–1825, https://doi.org/10.1175/1520-0469(1989)046<1815:AASOTS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Das Gupta, A. J., A. S. V. Murthy, and R. S. Nanjundiah, 2015: A study of singularities in the inviscid linear theory of sea breezes. Int. J. Adv. Eng. Sci. Appl. Math., 7, 33–37, https://doi.org/10.1007/s12572-015-0129-y.

    • Search Google Scholar
    • Export Citation

  • Drobinski, P., R. Rotunno, and T. Dubos, 2011: Linear theory of the sea breeze in a thermal wind. Quart. J. Roy. Meteor. Soc., 137, 1602–1609, https://doi.org/10.1002/qj.847.

    • Search Google Scholar
    • Export Citation

  • Du, Y., and R. Rotunno, 2015: Thermally driven diurnally periodic wind signals off the east coast of China. J. Atmos. Sci., 72, 2806–2821, https://doi.org/10.1175/JAS-D-14-0339.1.

    • Search Google Scholar
    • Export Citation

  • Du, Y., and R. Rotunno, 2018: Diurnal cycle of rainfall and winds near the south coast of China. J. Atmos. Sci., 75, 2065–2082, https://doi.org/10.1175/JAS-D-17-0397.1.

    • Search Google Scholar
    • Export Citation

  • Du, Y., R. Rotunno, and F. Zhang, 2019: Impact of vertical wind shear on gravity wave propagation in the land–sea-breeze circulation at the equator. J. Atmos. Sci., 76, 3247–3265, https://doi.org/10.1175/JAS-D-19-0069.1.

    • Search Google Scholar
    • Export Citation

  • Feng, S., Y. Fu, and Q. Xiao, 2012: Trends in the global tropopause thickness revealed by radiosondes. Geophys. Res. Lett., 39, L20706, https://doi.org/10.1029/2012GL053460.

    • Search Google Scholar
    • Export Citation

  • Fueglistaler, S., A. E. Dessler, T. J. Dunkerton, I. Folkins, Q. Fu, and P. W. Mote, 2009: Tropical tropopause layer. Rev. Geophys., 47, RG1004, https://doi.org/10.1029/2008RG000267.

    • Search Google Scholar
    • Export Citation

  • Gille, S. T., S. G. L. Smith, and N. M. Statom, 2005: Global observations of the land breeze. Geophys. Res. Lett., 32, L05605, https://doi.org/10.1029/2004GL022139.

    • Search Google Scholar
    • Export Citation

  • Hankinson, M. C. N., M. J. Reeder, and T. P. Lane, 2014: Gravity waves generated by convection during TWP-ICE: I. Inertia-gravity waves. J. Geophys. Res. Atmos., 119, 5269–5282, https://doi.org/10.1002/2013JD020724.

    • Search Google Scholar
    • Export Citation

  • Hecht, J. H., S. Kovalam, P. T. May, G. Mills, R. A. Vincent, R. L. Walterscheid, and J. Woithe, 2004: Airglow imager observations of atmospheric gravity waves at Alice Springs and Adelaide, Australia during the Darwin Area Wave Experiment (DAWEX). J. Geophys. Res., 109, D20S05, https://doi.org/10.1029/2004JD004697.

    • Search Google Scholar
    • Export Citation

  • Hu, S., and G. K. Vallis, 2019: Meridional structure and future changes of tropopause height and temperature. Quart. J. Roy. Meteor. Soc., 145, 2698–2717, https://doi.org/10.1002/qj.3587.

    • Search Google Scholar
    • Export Citation

  • Jiang, Q., 2012a: A linear theory of three-dimensional land–sea breezes. J. Atmos. Sci., 69, 1890–1909, https://doi.org/10.1175/JAS-D-11-0137.1.

    • Search Google Scholar
    • Export Citation

  • Jiang, Q., 2012b: On offshore propagating diurnal waves. J. Atmos. Sci., 69, 1562–1581, https://doi.org/10.1175/JAS-D-11-0220.1.

  • Kilpatrick, T., S.-P. Xie, and T. Nasuno, 2017: Diurnal convection-wind coupling in the Bay of Bengal. J. Geophys. Res. Atmos., 122, 9705–9720, https://doi.org/10.1002/2017JD027271.

    • Search Google Scholar
    • Export Citation

  • Lane, T. P., 2021: Does lower-stratospheric shear influence the mesoscale organization of convection? Geophys. Res. Lett., 48, e2020GL091025, https://doi.org/10.1029/2020GL091025.

    • Search Google Scholar
    • Export Citation

  • Li, Y., and J. Chao, 2016: An analytical solution for three-dimensional sea–land breeze. J. Atmos. Sci., 73, 41–54, https://doi.org/10.1175/JAS-D-14-0329.1.

    • Search Google Scholar
    • Export Citation

  • Lin, P., D. Paynter, Y. Ming, and V. Ramaswamy, 2017: Changes of the tropical tropopause layer under global warming. J. Climate, 30, 1245–1258, https://doi.org/10.1175/JCLI-D-16-0457.1.

    • Search Google Scholar
    • Export Citation

  • Lindzen, R. S., and K.-K. Tung, 1976: Banded convective activity and ducted gravity waves. Mon. Wea. Rev., 104, 1602–1617, https://doi.org/10.1175/1520-0493(1976)104<1602:BCAADG>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Mapes, B. E., T. T. Warner, and M. Xu, 2003: Diurnal patterns of rainfall in northwestern South America. Part III: Diurnal gravity waves and nocturnal convection offshore. Mon. Wea. Rev., 131, 830–844, https://doi.org/10.1175/1520-0493(2003)131<0830:DPORIN>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Minobe, S., and S. Takebayashi, 2015: Diurnal precipitation and high cloud frequency variability over the Gulf Stream and over the Kuroshio. Climate Dyn., 44, 2079–2095, https://doi.org/10.1007/s00382-014-2245-y.

    • Search Google Scholar
    • Export Citation

  • Mori, S., and Coauthors, 2004: Diurnal land–sea rainfall peak migration over Sumatera Island, Indonesian Maritime Continent, observed by TRMM satellite and intensive rawinsonde soundings. Mon. Wea. Rev., 132, 2021–2039, https://doi.org/10.1175/1520-0493(2004)132<2021:DLRPMO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Mowbray, D. E., and B. S. H. Rarity, 1967: A theoretical and experimental investigation of the phase configuration of internal waves of small amplitude in a density stratified liquid. J. Fluid Mech., 28, 1–16, https://doi.org/10.1017/S0022112067001867.

    • Search Google Scholar
    • Export Citation

  • Niino, H., 1987: The linear theory of land and sea breeze circulation. J. Meteor. Soc. Japan, 65, 901–921, https://doi.org/10.2151/jmsj1965.65.6_901.

    • Search Google Scholar
    • Export Citation

  • Ogura, Y., and N. A. Phillips, 1962: Scale analysis of deep and shallow convection in the atmosphere. J. Atmos. Sci., 19, 173–179, https://doi.org/10.1175/1520-0469(1962)019<0173:SAODAS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Pan, L. L., W. J. Randel, B. L. Gary, M. J. Mahoney, and E. J. Hintsa, 2004: Definitions and sharpness of the extratropical tropopause: A trace gas perspective. J. Geophys. Res., 109, D23103, https://doi.org/10.1029/2004JD004982.

    • Search Google Scholar
    • Export Citation

  • Pandya, R., D. Durran, and C. Bretherton, 1993: Comments on “Thermally forced gravity waves in an atmosphere at rest.” J. Atmos. Sci., 50, 4097–4101, https://doi.org/10.1175/1520-0469(1993)050<4097:COFGWI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Pandya, R., D. Durran, and M. L. Weisman, 2000: The influence of convective thermal forcing on the three-dimensional circulation around squall lines. J. Atmos. Sci., 57, 29–45, https://doi.org/10.1175/1520-0469(2000)057<0029:TIOCTF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Protat, A., and I. McRobert, 2020a: RV Investigator IN2019_v06 BASTA radar and radiosonde data. Commonwealth Scientific and Industrial Research Organisation, accessed 23 February 2023, https://doi.org/10.25919/5ef5a70ae2be9.

  • Protat, A., and I. McRobert, 2020b: Three-dimensional wind profiles using a stabilized shipborne cloud radar in wind profiler mode. Atmos. Meas. Tech., 13, 3609–3620, https://doi.org/10.5194/amt-13-3609-2020.

    • Search Google Scholar
    • Export Citation

  • Protat, A., V. Louf, J. Soderholm, J. Brook, and W. Ponsonby, 2022: Three-way calibration checks using ground-based, ship-based, and spaceborne radars. Atmos. Meas. Tech., 15, 915–926, https://doi.org/10.5194/amt-15-915-2022.

    • Search Google Scholar
    • Export Citation

  • Qian, T., C. C. Epifanio, and F. Zhang, 2009: Linear theory calculations for the sea breeze in a background wind: The equatorial case. J. Atmos. Sci., 66, 1749–1763, https://doi.org/10.1175/2008JAS2851.1.

    • Search Google Scholar
    • Export Citation

  • Qian, T., C. C. Epifanio, and F. Zhang, 2012: Topographic effects on the tropical land and sea breeze. J. Atmos. Sci., 69, 130–149, https://doi.org/10.1175/JAS-D-11-011.1.

    • Search Google Scholar
    • Export Citation

  • Robinson, F. J., S. C. Sherwood, and Y. Li, 2008: Resonant response of deep convection to surface hot spots. J. Atmos. Sci., 65, 276–286, https://doi.org/10.1175/2007JAS2398.1.

    • Search Google Scholar
    • Export Citation

  • Rotunno, R., 1983: On the linear theory of the land and sea breeze. J. Atmos. Sci., 40, 1999–2009, https://doi.org/10.1175/1520-0469(1983)040<1999:OTLTOT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Schmidt, T., G. Beyerle, S. Heise, J. Wickert, and M. Rothacher, 2006: A climatology of multiple tropopauses derived from GPS radio occultations with CHAMP and SAC-C. Geophys. Res. Lett., 33, L04808, https://doi.org/10.1029/2005GL024600.

    • Search Google Scholar
    • Export Citation

  • Short, E., 2023: rotunno83. GitHub, accessed 17 March 2023, https://github.com/eshort0401/rotunno83.

  • Sutherland, B. R., 2010: Internal Gravity Waves. Cambridge University Press, 377 pp., https://doi.org/10.1017/CBO9780511780318.

  • Vincent, C. L., and T. P. Lane, 2016: Evolution of the diurnal precipitation cycle with the passage of a Madden–Julian oscillation event through the Maritime Continent. Mon. Wea. Rev., 144, 1983–2005, https://doi.org/10.1175/MWR-D-15-0326.1.

    • Search Google Scholar
    • Export Citation

  • Vincent, C. L., and T. P. Lane, 2018: Mesoscale variation in diabatic heating around Sumatra, and its modulation with the Madden–Julian oscillation. Mon. Wea. Rev., 146, 2599–2614, https://doi.org/10.1175/MWR-D-17-0392.1.

    • Search Google Scholar
    • Export Citation

  • Vincent, R. A., A. MacKinnon, I. M. Reid, and M. J. Alexander, 2004: VHF profiler observations of winds and waves in the troposphere during the Darwin Area Wave Experiment (DAWEX). J. Geophys. Res., 109, D20S02, https://doi.org/10.1029/2004JD004714.

    • Search Google Scholar
    • Export Citation

  • Whittaker, E. T., and G. N. Watson, 1996: A Course of Modern Analysis. 4th ed. Cambridge Mathematical Library, Cambridge University Press, 608 pp., https://doi.org/10.1017/CBO9780511608759.

  • Wünsche, A., 2003: Generalized Hermite polynomials associated with functions of parabolic cylinder. Appl. Math. Comput., 141, 197–213, https://doi.org/10.1016/S0096-3003(02)00333-8.

    • Search Google Scholar
    • Export Citation

  • Yan, H., and R. A. Anthes, 1987: The effect of latitude on the sea breeze. Mon. Wea. Rev., 115, 936–956, https://doi.org/10.1175/1520-0493(1987)115<0936:TEOLOT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Yang, G.-Y., and J. Slingo, 2001: The diurnal cycle in the tropics. Mon. Wea. Rev., 129, 784–801, https://doi.org/10.1175/1520-0493(2001)129<0784:TDCITT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Yoneyama, K., and C. Zhang, 2020: Years of the Maritime Continent. Geophys. Res. Lett., 47, e2020GL087182, https://doi.org/10.1029/2020GL087182.

    • Search Google Scholar
    • Export Citation

Supplementary Materials

Save
Cover Journal of the Atmospheric Sciences
Journal of the Atmospheric Sciences

  • Abramowitz, M., and I. A. Stegun, 1972: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, 1076 pp.

  • Alexander, M. J., P. T. May, and J. H. Beres, 2004: Gravity waves generated by convection in the Darwin area during the Darwin Area Wave Experiment. J. Geophys. Res., 109, D20S04, https://doi.org/10.1029/2004JD004729.

    • Search Google Scholar
    • Export Citation

  • Bergemann, M., C. Jakob, and T. P. Lane, 2015: Global detection and analysis of coastline-associated rainfall using an objective pattern recognition technique. J. Climate, 28, 7225–7236, https://doi.org/10.1175/JCLI-D-15-0098.1.

    • Search Google Scholar
    • Export Citation

  • Dacie, S., and Coauthors, 2019: A 1D RCE study of factors affecting the tropical tropopause layer and surface climate. J. Climate, 32, 6769–6782, https://doi.org/10.1175/JCLI-D-18-0778.1.

    • Search Google Scholar
    • Export Citation

  • Dalu, G. A., and R. A. Pielke, 1989: An analytical study of the sea breeze. J. Atmos. Sci., 46, 1815–1825, https://doi.org/10.1175/1520-0469(1989)046<1815:AASOTS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Das Gupta, A. J., A. S. V. Murthy, and R. S. Nanjundiah, 2015: A study of singularities in the inviscid linear theory of sea breezes. Int. J. Adv. Eng. Sci. Appl. Math., 7, 33–37, https://doi.org/10.1007/s12572-015-0129-y.

    • Search Google Scholar
    • Export Citation

  • Drobinski, P., R. Rotunno, and T. Dubos, 2011: Linear theory of the sea breeze in a thermal wind. Quart. J. Roy. Meteor. Soc., 137, 1602–1609, https://doi.org/10.1002/qj.847.

    • Search Google Scholar
    • Export Citation

  • Du, Y., and R. Rotunno, 2015: Thermally driven diurnally periodic wind signals off the east coast of China. J. Atmos. Sci., 72, 2806–2821, https://doi.org/10.1175/JAS-D-14-0339.1.

    • Search Google Scholar
    • Export Citation

  • Du, Y., and R. Rotunno, 2018: Diurnal cycle of rainfall and winds near the south coast of China. J. Atmos. Sci., 75, 2065–2082, https://doi.org/10.1175/JAS-D-17-0397.1.

    • Search Google Scholar
    • Export Citation

  • Du, Y., R. Rotunno, and F. Zhang, 2019: Impact of vertical wind shear on gravity wave propagation in the land–sea-breeze circulation at the equator. J. Atmos. Sci., 76, 3247–3265, https://doi.org/10.1175/JAS-D-19-0069.1.

    • Search Google Scholar
    • Export Citation

  • Feng, S., Y. Fu, and Q. Xiao, 2012: Trends in the global tropopause thickness revealed by radiosondes. Geophys. Res. Lett., 39, L20706, https://doi.org/10.1029/2012GL053460.

    • Search Google Scholar
    • Export Citation

  • Fueglistaler, S., A. E. Dessler, T. J. Dunkerton, I. Folkins, Q. Fu, and P. W. Mote, 2009: Tropical tropopause layer. Rev. Geophys., 47, RG1004, https://doi.org/10.1029/2008RG000267.

    • Search Google Scholar
    • Export Citation

  • Gille, S. T., S. G. L. Smith, and N. M. Statom, 2005: Global observations of the land breeze. Geophys. Res. Lett., 32, L05605, https://doi.org/10.1029/2004GL022139.

    • Search Google Scholar
    • Export Citation

  • Hankinson, M. C. N., M. J. Reeder, and T. P. Lane, 2014: Gravity waves generated by convection during TWP-ICE: I. Inertia-gravity waves. J. Geophys. Res. Atmos., 119, 5269–5282, https://doi.org/10.1002/2013JD020724.

    • Search Google Scholar
    • Export Citation

  • Hecht, J. H., S. Kovalam, P. T. May, G. Mills, R. A. Vincent, R. L. Walterscheid, and J. Woithe, 2004: Airglow imager observations of atmospheric gravity waves at Alice Springs and Adelaide, Australia during the Darwin Area Wave Experiment (DAWEX). J. Geophys. Res., 109, D20S05, https://doi.org/10.1029/2004JD004697.

    • Search Google Scholar
    • Export Citation

  • Hu, S., and G. K. Vallis, 2019: Meridional structure and future changes of tropopause height and temperature. Quart. J. Roy. Meteor. Soc., 145, 2698–2717, https://doi.org/10.1002/qj.3587.

    • Search Google Scholar
    • Export Citation

  • Jiang, Q., 2012a: A linear theory of three-dimensional land–sea breezes. J. Atmos. Sci., 69, 1890–1909, https://doi.org/10.1175/JAS-D-11-0137.1.

    • Search Google Scholar
    • Export Citation

  • Jiang, Q., 2012b: On offshore propagating diurnal waves. J. Atmos. Sci., 69, 1562–1581, https://doi.org/10.1175/JAS-D-11-0220.1.

  • Kilpatrick, T., S.-P. Xie, and T. Nasuno, 2017: Diurnal convection-wind coupling in the Bay of Bengal. J. Geophys. Res. Atmos., 122, 9705–9720, https://doi.org/10.1002/2017JD027271.

    • Search Google Scholar
    • Export Citation

  • Lane, T. P., 2021: Does lower-stratospheric shear influence the mesoscale organization of convection? Geophys. Res. Lett., 48, e2020GL091025, https://doi.org/10.1029/2020GL091025.

    • Search Google Scholar
    • Export Citation

  • Li, Y., and J. Chao, 2016: An analytical solution for three-dimensional sea–land breeze. J. Atmos. Sci., 73, 41–54, https://doi.org/10.1175/JAS-D-14-0329.1.

    • Search Google Scholar
    • Export Citation

  • Lin, P., D. Paynter, Y. Ming, and V. Ramaswamy, 2017: Changes of the tropical tropopause layer under global warming. J. Climate, 30, 1245–1258, https://doi.org/10.1175/JCLI-D-16-0457.1.

    • Search Google Scholar
    • Export Citation

  • Lindzen, R. S., and K.-K. Tung, 1976: Banded convective activity and ducted gravity waves. Mon. Wea. Rev., 104, 1602–1617, https://doi.org/10.1175/1520-0493(1976)104<1602:BCAADG>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Mapes, B. E., T. T. Warner, and M. Xu, 2003: Diurnal patterns of rainfall in northwestern South America. Part III: Diurnal gravity waves and nocturnal convection offshore. Mon. Wea. Rev., 131, 830–844, https://doi.org/10.1175/1520-0493(2003)131<0830:DPORIN>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Minobe, S., and S. Takebayashi, 2015: Diurnal precipitation and high cloud frequency variability over the Gulf Stream and over the Kuroshio. Climate Dyn., 44, 2079–2095, https://doi.org/10.1007/s00382-014-2245-y.

    • Search Google Scholar
    • Export Citation

  • Mori, S., and Coauthors, 2004: Diurnal land–sea rainfall peak migration over Sumatera Island, Indonesian Maritime Continent, observed by TRMM satellite and intensive rawinsonde soundings. Mon. Wea. Rev., 132, 2021–2039, https://doi.org/10.1175/1520-0493(2004)132<2021:DLRPMO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Mowbray, D. E., and B. S. H. Rarity, 1967: A theoretical and experimental investigation of the phase configuration of internal waves of small amplitude in a density stratified liquid. J. Fluid Mech., 28, 1–16, https://doi.org/10.1017/S0022112067001867.

    • Search Google Scholar
    • Export Citation

  • Niino, H., 1987: The linear theory of land and sea breeze circulation. J. Meteor. Soc. Japan, 65, 901–921, https://doi.org/10.2151/jmsj1965.65.6_901.

    • Search Google Scholar
    • Export Citation

  • Ogura, Y., and N. A. Phillips, 1962: Scale analysis of deep and shallow convection in the atmosphere. J. Atmos. Sci., 19, 173–179, https://doi.org/10.1175/1520-0469(1962)019<0173:SAODAS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Pan, L. L., W. J. Randel, B. L. Gary, M. J. Mahoney, and E. J. Hintsa, 2004: Definitions and sharpness of the extratropical tropopause: A trace gas perspective. J. Geophys. Res., 109, D23103, https://doi.org/10.1029/2004JD004982.

    • Search Google Scholar
    • Export Citation

  • Pandya, R., D. Durran, and C. Bretherton, 1993: Comments on “Thermally forced gravity waves in an atmosphere at rest.” J. Atmos. Sci., 50, 4097–4101, https://doi.org/10.1175/1520-0469(1993)050<4097:COFGWI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Pandya, R., D. Durran, and M. L. Weisman, 2000: The influence of convective thermal forcing on the three-dimensional circulation around squall lines. J. Atmos. Sci., 57, 29–45, https://doi.org/10.1175/1520-0469(2000)057<0029:TIOCTF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Protat, A., and I. McRobert, 2020a: RV Investigator IN2019_v06 BASTA radar and radiosonde data. Commonwealth Scientific and Industrial Research Organisation, accessed 23 February 2023, https://doi.org/10.25919/5ef5a70ae2be9.

  • Protat, A., and I. McRobert, 2020b: Three-dimensional wind profiles using a stabilized shipborne cloud radar in wind profiler mode. Atmos. Meas. Tech., 13, 3609–3620, https://doi.org/10.5194/amt-13-3609-2020.

    • Search Google Scholar
    • Export Citation

  • Protat, A., V. Louf, J. Soderholm, J. Brook, and W. Ponsonby, 2022: Three-way calibration checks using ground-based, ship-based, and spaceborne radars. Atmos. Meas. Tech., 15, 915–926, https://doi.org/10.5194/amt-15-915-2022.

    • Search Google Scholar
    • Export Citation

  • Qian, T., C. C. Epifanio, and F. Zhang, 2009: Linear theory calculations for the sea breeze in a background wind: The equatorial case. J. Atmos. Sci., 66, 1749–1763, https://doi.org/10.1175/2008JAS2851.1.

    • Search Google Scholar
    • Export Citation

  • Qian, T., C. C. Epifanio, and F. Zhang, 2012: Topographic effects on the tropical land and sea breeze. J. Atmos. Sci., 69, 130–149, https://doi.org/10.1175/JAS-D-11-011.1.

    • Search Google Scholar
    • Export Citation

  • Robinson, F. J., S. C. Sherwood, and Y. Li, 2008: Resonant response of deep convection to surface hot spots. J. Atmos. Sci., 65, 276–286, https://doi.org/10.1175/2007JAS2398.1.

    • Search Google Scholar
    • Export Citation

  • Rotunno, R., 1983: On the linear theory of the land and sea breeze. J. Atmos. Sci., 40, 1999–2009, https://doi.org/10.1175/1520-0469(1983)040<1999:OTLTOT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Schmidt, T., G. Beyerle, S. Heise, J. Wickert, and M. Rothacher, 2006: A climatology of multiple tropopauses derived from GPS radio occultations with CHAMP and SAC-C. Geophys. Res. Lett., 33, L04808, https://doi.org/10.1029/2005GL024600.

    • Search Google Scholar
    • Export Citation

  • Short, E., 2023: rotunno83. GitHub, accessed 17 March 2023, https://github.com/eshort0401/rotunno83.

  • Sutherland, B. R., 2010: Internal Gravity Waves. Cambridge University Press, 377 pp., https://doi.org/10.1017/CBO9780511780318.

  • Vincent, C. L., and T. P. Lane, 2016: Evolution of the diurnal precipitation cycle with the passage of a Madden–Julian oscillation event through the Maritime Continent. Mon. Wea. Rev., 144, 1983–2005, https://doi.org/10.1175/MWR-D-15-0326.1.

    • Search Google Scholar
    • Export Citation

  • Vincent, C. L., and T. P. Lane, 2018: Mesoscale variation in diabatic heating around Sumatra, and its modulation with the Madden–Julian oscillation. Mon. Wea. Rev., 146, 2599–2614, https://doi.org/10.1175/MWR-D-17-0392.1.

    • Search Google Scholar
    • Export Citation

  • Vincent, R. A., A. MacKinnon, I. M. Reid, and M. J. Alexander, 2004: VHF profiler observations of winds and waves in the troposphere during the Darwin Area Wave Experiment (DAWEX). J. Geophys. Res., 109, D20S02, https://doi.org/10.1029/2004JD004714.

    • Search Google Scholar
    • Export Citation

  • Whittaker, E. T., and G. N. Watson, 1996: A Course of Modern Analysis. 4th ed. Cambridge Mathematical Library, Cambridge University Press, 608 pp., https://doi.org/10.1017/CBO9780511608759.

  • Wünsche, A., 2003: Generalized Hermite polynomials associated with functions of parabolic cylinder. Appl. Math. Comput., 141, 197–213, https://doi.org/10.1016/S0096-3003(02)00333-8.

    • Search Google Scholar
    • Export Citation

  • Yan, H., and R. A. Anthes, 1987: The effect of latitude on the sea breeze. Mon. Wea. Rev., 115, 936–956, https://doi.org/10.1175/1520-0493(1987)115<0936:TEOLOT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Yang, G.-Y., and J. Slingo, 2001: The diurnal cycle in the tropics. Mon. Wea. Rev., 129, 784–801, https://doi.org/10.1175/1520-0493(2001)129<0784:TDCITT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Yoneyama, K., and C. Zhang, 2020: Years of the Maritime Continent. Geophys. Res. Lett., 47, e2020GL087182, https://doi.org/10.1029/2020GL087182.

    • Search Google Scholar
    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    The heating function given by Eq. (6) at 1200 LST, i.e., t*=0, with L = 50 km, H = 1 km, and Q0 = 1.2 × 10−5 m s−3.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    Coastline perpendicular vertical cross sections of the piecewise-constant solution for the coastline perpendicular horizontal winds in dimensional coordinates u* at, (a) 1200, (b) 1500, (c) 1800, and (d) 2100 LST, noting that these times correspond to values of 0, 3, 6, and 9 h for t*, respectively. The horizontal coordinate x* gives distance from the coastline. The coastal width is L = 50 km, the heating depth H = 1 km, and Brunt–Väisälä frequencies are N1 = 0.01 s−1 and N2 = 0.03 s−1, giving N=3. The change in stability occurs at z*=H1=2km, as indicated by the horizontal dotted line. The heating amplitude is Q0 = 1.2 × 10−5 m s−3.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    As in Fig. 2, but for (a) the horizontal velocities u* resulting from heating F1 below the stability change at H1 = 2 km and (b) heating F2 above H1. Note the difference in color-bar scales, with the overall solution depicted in Fig. 2 determined almost entirely by F1.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    As in Fig. 2, but for (a) the vertical velocity in dimensional coordinates w* and (b) the velocity field in dimensional coordinates (u*, w*), with arrows and shading illustrating the direction and magnitude of the velocity vectors.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    As in Fig. 2, but for the transition-layer solution for the coastline-perpendicular horizontal winds u*. The transition layer begins at z*=H1=1.5km and ends at z*=H2=2.5km, as depicted by the horizontal dotted lines, with N1 = 0.01 s−1 and N2 = 0.03 s−1, so that N=3, as before.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    As in Fig. 5, but for (a) the horizontal velocities u* resulting from heating F1 below H1 = 1.5 km, (b) heating FTL within the transition layer between H1 = 1.5 km and H2 = 2.5 km, and (c) heating F2 above H2. Note the difference in color-bar scales, with the overall solution depicted in Fig. 5 determined almost entirely by F1.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    As in Fig. 4, but for the transition-layer solutions for (a) w* and (b) the velocity field (u*, w*).

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    Plots illustrating how (a) the reflection coefficient r1, (b) the refraction coefficient r2, and (c),(d) the ducting coefficient r3, change as the transition layer deepens, for various values of the nondimensional vertical wavenumber m, noting m=m*H, where m* is the wavenumber in dimensional coordinates, and H is the heating depth. The transition layer begins at z=H1=2, with N=3. Dashed gray lines show the large H2−H1 limit values 0, 1/N, and N for the coefficients r1, r2, and r3, respectively.

  • <sc>Fig</sc>. 9.
    View in gallery
    Fig. 9.

    Maps depicting, (a) Australia’s Northern Territory, its capital Darwin, and surrounding landmarks, and (b) the region within the red dashed box in (a), showing the location of the R/V Investigator between the 20 Nov and the 2 Dec 2019.

  • <sc>Fig</sc>. 10.
    View in gallery
    Fig. 10.

    The vertical profile of the Brunt–Väisälä frequency N from the soundings obtained during the portion of the R/V Investigator voyage depicted in Fig. 9b. The light gray line depicts the Brunt–Väisälä frequency at 0000 LST 20 Nov 2019, the dark gray line depicts the average over 20 Nov 2019, while the thicker black line depicts the average over the entire 20 Nov–2 Dec 2019 period, with a 200 m running mean applied to this average to further smooth small-scale noise.

  • <sc>Fig</sc>. 11.
    View in gallery
    Fig. 11.

    Hovmöller diagrams of (a) the meridional winds, (b) the meridional wind perturbations against a 24 h centered-running-mean background wind, and (c) the average perturbations at each time of day, from the soundings taken on board the R/V Investigator between 20 Nov and 2 Dec 2019. The location of the R/V Investigator during this period is depicted in Fig. 9b.

  • <sc>Fig</sc>. 12.
    View in gallery
    Fig. 12.

    The heating function given by Eq. (55) at 1200 LST, i.e., t*=0, with L = 100 km, H = 12 km, D = 4 km, and Q0 = 6 × 10−6 m s−3.

  • <sc>Fig</sc>. 13.
    View in gallery
    Fig. 13.

    The piecewise-constant solution for the convective-line-perpendicular horizontal winds u*, using the heating function given by Eq. (55), with H1 = 17 km, L = 100 km, D = 4 km, H = 12 km, and Q0 = 6 × 10−6 m s−3. The Brunt–Väisälä frequencies are N1 = 0.01 s−1 and N2 = 0.025 s−1, so that N=2.5, and the f plane is chosen based on a constant latitude of 11.5°S. Depicted are the winds at, (a) 1200, (b) 1500, (c) 1800, and (d) 2100 LST. The vertical gray dashed lines at x*=−600,−800, and−1000km show the vertical transects considered in Fig. 17.

  • <sc>Fig</sc>. 14.
    View in gallery
    Fig. 14.

    As in Fig. 13, but for (a) the horizontal velocities u* at 1500 LST resulting from heating F1 below the stability change at H1 = 17 km, as depicted by the horizontal dotted line, and (b) heating F2 above H1. Note the difference in color-bar scales, with the overall solution depicted in Fig. 13 determined mostly by F1.

  • <sc>Fig</sc>. 15.
    View in gallery
    Fig. 15.

    As in Fig. 13, but for the transition-layer solution, with H1 = 15 km and H2 = 19 km, as depicted by the horizontal dotted lines.

  • <sc>Fig</sc>. 16.
    View in gallery
    Fig. 16.

    As in Fig. 15, but for the horizontal velocities u* at 1500 LST resulting from (a) heating F1 below H1 = 15 km, (b) heating FTL within the transition layer between H1 = 15 km and H2 = 19 km, and (c) heating F2 above H2. Note the difference in color-bar scales, with the overall solution depicted in Fig. 15 determined mostly by F1 and FTL.

  • <sc>Fig</sc>. 17.
    View in gallery
    Fig. 17.

    Hovmöller diagrams of the convective-line-perpendicular horizontal winds u* from (a),(c),(e) the piecewise-constant solution and (b),(d),(e) the transition-layer solutions depicted in Figs. 13 and 15 along the gray dashed vertical transects at (a),(b) x*=−600km, (c),(d) x*=−800km, and (e),(f) x*=−1000 km. The heights of the stability changes H1 and H2 are depicted by the horizontal dotted lines.

Fig. 1.

The heating function given by Eq. (6) at 1200 LST, i.e., t*=0, with L = 50 km, H = 1 km, and Q0 = 1.2 × 10−5 m s−3.
Fig. 2.

Coastline perpendicular vertical cross sections of the piecewise-constant solution for the coastline perpendicular horizontal winds in dimensional coordinates u* at, (a) 1200, (b) 1500, (c) 1800, and (d) 2100 LST, noting that these times correspond to values of 0, 3, 6, and 9 h for t*, respectively. The horizontal coordinate x* gives distance from the coastline. The coastal width is L = 50 km, the heating depth H = 1 km, and Brunt–Väisälä frequencies are N1 = 0.01 s−1 and N2 = 0.03 s−1, giving N=3. The change in stability occurs at z*=H1=2km, as indicated by the horizontal dotted line. The heating amplitude is Q0 = 1.2 × 10−5 m s−3.
Fig. 3.

As in Fig. 2, but for (a) the horizontal velocities u* resulting from heating F1 below the stability change at H1 = 2 km and (b) heating F2 above H1. Note the difference in color-bar scales, with the overall solution depicted in Fig. 2 determined almost entirely by F1.
Fig. 4.

As in Fig. 2, but for (a) the vertical velocity in dimensional coordinates w* and (b) the velocity field in dimensional coordinates (u*, w*), with arrows and shading illustrating the direction and magnitude of the velocity vectors.
Fig. 5.

As in Fig. 2, but for the transition-layer solution for the coastline-perpendicular horizontal winds u*. The transition layer begins at z*=H1=1.5km and ends at z*=H2=2.5km, as depicted by the horizontal dotted lines, with N1 = 0.01 s−1 and N2 = 0.03 s−1, so that N=3, as before.
Fig. 6.

As in Fig. 5, but for (a) the horizontal velocities u* resulting from heating F1 below H1 = 1.5 km, (b) heating FTL within the transition layer between H1 = 1.5 km and H2 = 2.5 km, and (c) heating F2 above H2. Note the difference in color-bar scales, with the overall solution depicted in Fig. 5 determined almost entirely by F1.
Fig. 7.

As in Fig. 4, but for the transition-layer solutions for (a) w* and (b) the velocity field (u*, w*).
Fig. 8.

Plots illustrating how (a) the reflection coefficient r1, (b) the refraction coefficient r2, and (c),(d) the ducting coefficient r3, change as the transition layer deepens, for various values of the nondimensional vertical wavenumber m, noting m=m*H, where m* is the wavenumber in dimensional coordinates, and H is the heating depth. The transition layer begins at z=H1=2, with N=3. Dashed gray lines show the large H2−H1 limit values 0, 1/N, and N for the coefficients r1, r2, and r3, respectively.
Fig. 9.

Maps depicting, (a) Australia’s Northern Territory, its capital Darwin, and surrounding landmarks, and (b) the region within the red dashed box in (a), showing the location of the R/V Investigator between the 20 Nov and the 2 Dec 2019.
Fig. 10.

The vertical profile of the Brunt–Väisälä frequency N from the soundings obtained during the portion of the R/V Investigator voyage depicted in Fig. 9b. The light gray line depicts the Brunt–Väisälä frequency at 0000 LST 20 Nov 2019, the dark gray line depicts the average over 20 Nov 2019, while the thicker black line depicts the average over the entire 20 Nov–2 Dec 2019 period, with a 200 m running mean applied to this average to further smooth small-scale noise.
Fig. 11.

Hovmöller diagrams of (a) the meridional winds, (b) the meridional wind perturbations against a 24 h centered-running-mean background wind, and (c) the average perturbations at each time of day, from the soundings taken on board the R/V Investigator between 20 Nov and 2 Dec 2019. The location of the R/V Investigator during this period is depicted in Fig. 9b.
Fig. 12.

The heating function given by Eq. (55) at 1200 LST, i.e., t*=0, with L = 100 km, H = 12 km, D = 4 km, and Q0 = 6 × 10−6 m s−3.
Fig. 13.

The piecewise-constant solution for the convective-line-perpendicular horizontal winds u*, using the heating function given by Eq. (55), with H1 = 17 km, L = 100 km, D = 4 km, H = 12 km, and Q0 = 6 × 10−6 m s−3. The Brunt–Väisälä frequencies are N1 = 0.01 s−1 and N2 = 0.025 s−1, so that N=2.5, and the f plane is chosen based on a constant latitude of 11.5°S. Depicted are the winds at, (a) 1200, (b) 1500, (c) 1800, and (d) 2100 LST. The vertical gray dashed lines at x*=−600,−800, and−1000km show the vertical transects considered in Fig. 17.
Fig. 14.

As in Fig. 13, but for (a) the horizontal velocities u* at 1500 LST resulting from heating F1 below the stability change at H1 = 17 km, as depicted by the horizontal dotted line, and (b) heating F2 above H1. Note the difference in color-bar scales, with the overall solution depicted in Fig. 13 determined mostly by F1.
Fig. 15.

As in Fig. 13, but for the transition-layer solution, with H1 = 15 km and H2 = 19 km, as depicted by the horizontal dotted lines.
Fig. 16.

As in Fig. 15, but for the horizontal velocities u* at 1500 LST resulting from (a) heating F1 below H1 = 15 km, (b) heating FTL within the transition layer between H1 = 15 km and H2 = 19 km, and (c) heating F2 above H2. Note the difference in color-bar scales, with the overall solution depicted in Fig. 15 determined mostly by F1 and FTL.
Fig. 17.

Hovmöller diagrams of the convective-line-perpendicular horizontal winds u* from (a),(c),(e) the piecewise-constant solution and (b),(d),(e) the transition-layer solutions depicted in Figs. 13 and 15 along the gray dashed vertical transects at (a),(b) x*=−600km, (c),(d) x*=−800km, and (e),(f) x*=−1000 km. The heights of the stability changes H1 and H2 are depicted by the horizontal dotted lines.
All Time Past Year Past 30 Days
Abstract Views 1099 256 0
Full Text Views 13173 12961 316
PDF Downloads 434 179 9