## 1. Introduction

Over the past decade, wavelike diurnal variations in surface winds and surface wind divergence, stratocumulus, and 800-hPa temperature anomaly propagating offshore of tropical and subtropical continents have been documented by several numerical modeling and satellite observation studies. Yang and Slingo (2001) analyzed the diurnal cycle of convection, cloudiness, and surface temperature in the tropics using brightness temperature from multiple satellites and found that a strong diurnal signal over land can propagate several hundreds of kilometers over adjacent oceans. Mapes et al. (2003) examined diurnal variations of 800-hPa temperature anomalies over the tropical area of the eastern Pacific using mesoscale model simulations. Diurnal wavelike temperature anomalies were found between 0°–3°S and 5°–7°N and were believed to have played a role in triggering offshore convection systems. These signals propagated offshore at approximately 15 m s^{−1} while slowly decaying and were still detectible at least 700 km away from the coastline. Diurnal cycles of liquid water path and low cloud amount over tropical and subtropical oceans derived from satellite data were reported by Rozendaal et al. (1995) and Wood et al. (2002). Gille et al. (2005) used 4-times-daily sea surface winds from the Quick Scatterometer (QuikSCAT) and Advanced Earth Observing Satellite II (ADEOS-II) tandem scatterometer mission to study diurnal variation of global land and sea breezes and found that diurnal perturbations of sea surface winds propagate progressively offshore at speeds ranging from 2 to 15 m s^{−1}, resembling gravity waves. These perturbations are detectible several hundred kilometers offshore, largely confined between the 30° latitudinal parallels, and in general more pronounced to the west than the east of continents.

Satellite observations indicated that, while diurnal variations of low-level winds and cloudiness are virtually ubiquitous over the tropical and subtropical oceans, a pronounced maximum exists over the subtropical southeast Pacific (e.g., Gille et al. 2005; Wood et al. 2009). Garreaud and Muñoz (2004) performed 15-day regional model simulations over this area and noticed that a band of upward motion formed regularly along the coast of southern Peru during afternoons and propagated southwestward offshore at approximately 1° h^{−1} (i.e., ~25–30 m s^{−1}). This band, referred to as an “upsidence wave” in Garreaud and Muñoz, was typically 5 km deep and 400 km wide and played a significant role in the diurnal variation of cloudiness over the southeast Pacific. Wood et al. computed diurnal cycle of surface divergence utilizing 4-times-daily satellite data for April–September 2003. A band of subsidence was evident in their analysis, which was detached from the southern Peru coast in local afternoons, propagated over southeastern Pacific at approximately 25 m s^{−1}, and was detectible at least 2000 km offshore. Similar waves were also identified by Muñoz (2008) using satellite data as well as numerical simulations. These offshore diurnal waves with a horizontal dimension of a few hundred kilometers and discernable over a large ocean area were believed to have a significant impact on the stratocumulus deck over the southeastern Pacific, which plays a nonnegligible role in the earth’s radiation budget. Because of their climatological importance, dynamics of the upsidence wave (or subsidence wave, referred to as diurnal waves in this study) and their interaction with the southeastern Pacific stratocumulus were cited as one of the objectives of the recently completed multiagency international field campaign, the Variability of the American Monsoon Systems (VAMOS) Ocean–Cloud–Atmosphere–Land Study (VOCALS) (Wood et al. 2011). Some of the observations from VOCALS provided further evidence for the existence of diurnal waves offshore of southern Peru and northern Chile (Rahn and Garreaud 2010). It is worth noting that there is also a body of literature on diurnal variation of precipitation over land (e.g., Carbone et al. 2002; Robinson et al. 2008; Li and Smith 2010), which might be dynamically relevant to the offshore diurnal waves examined in this study.

Diurnal waves associated with oscillating land–sea differential heating were first discussed by Sun and Orlanski (1981). Rotunno (1983, hereafter R83) examined the linear response of a uniform atmosphere to a specified diurnal heating source with emphasis on the latitudinal dependence of land–sea breezes. As described by R83, for diurnal perturbations a critical latitude,

For simplicity, most previous analytical studies assumed that winds and buoyancy frequencies are constant with height. However, the real atmosphere is often characterized by sharp vertical variations in winds and stratification. For example, over a tropical area the lower troposphere is typically less stable than the midtroposphere, and over a subtropical ocean the marine boundary layer is often capped by an inversion. The objective of this study is to examine characteristics of offshore diurnal waves induced by land–sea differential heating within an atmospheric boundary layer or a deep mixed layer, which typically develops over arid tropical (subtropical) continents and extends well above the marine boundary layer over adjacent oceans. The emphasis is on the marine boundary layer top inversion, vertical stability variation, steady background wind, and BL viscosity effects. The remainder of this paper is organized as follows. In section 2, the linear wave equations, wave trapping conditions, and governing parameters are discussed. Diurnal waves generated by heating within a boundary layer are examined in section 3. In section 4, waves associated with a deep heating source located above a frictional boundary layer are investigated. The results are summarized in section 5.

## 2. Linear theory and control parameters

*Q*(

*x*,

*y*,

*z*,

*t*) can be written as (e.g., Li and Smith 2010)

*U*and

*V*are the horizontal ambient wind components;

*u*,

*υ*, and

*w*are perturbation velocity components;

*f*is the Coriolis parameter;

*α*is Rayleigh friction coefficient; and

*Q*, with units of watts per cubic meter, is the maximum heating rate or vertical heat flux convergence; and

_{o}*q*(

*x*,

*y*) and

*Z*(

*z*) denote the nondimensional horizontal and vertical distributions of the heat source, respectively. The vertical heating profile

*Z*(

*z*) satisfies

*D*is the vertical dimension of the heating source.

*ϕ*represents

*u*,

*υ*,

*w*,

*p*′ or

*b*, and

*k*and

*l*denote the zonal and meridional wavenumbers, Eqs. (1a)–(1e) can be combined into a single wave equation in terms of

*q*(

*x*,

*y*). Once

For an atmosphere composed of discrete layers characterized by a uniform wind and constant buoyancy frequency in each layer, Eq. (2) can be solved for each layer with matching conditions applied at interfaces (appendix A). Two groups of solutions are examined in the following sections. In section 3, a two-dimensional diurnal heating source is confined in a shallow boundary layer capped by a sharp inversion, above which a deep uniformly stratified atmosphere is located. The objective is to understand the role of a marine boundary top inversion, steady large-scale winds, stability variation, and boundary layer viscosity effects on wave properties. Solutions with a 5-km-deep heating source, mimicking a deep well-mixed layer over the central Andes or Africa (e.g., Dunion and Velden 2004), are examined in section 4. We are particularly interested in the sensitivity of diurnal waves to vertical variations in stratification and background winds, and the role a frictional boundary layer plays in decaying offshore waves.

## 3. Boundary layer heating under an inversion

We first consider a two-layer atmosphere with a relatively shallow lower layer located below a sharp inversion *N _{j}*, uniform winds (

*U*,

_{j}*V*), and Rayleigh friction coefficients

_{j}*α*, where

_{j}*j*is the layer index. This two-layer system represents a shallow marine boundary layer separated from a deep atmosphere above by a marine boundary layer top inversion.

### a. Diurnal wave solutions in a two-layer atmosphere

*z*= −

*D*and the BL top at

*z*= 0, where

*D*is the BL depth. Solutions in these two layers can be written as

*j*for a hydrostatic wave. As there is no heating in the upper layer, the radiation top boundary condition requires

*B*

_{2}= 0. The constant coefficients,

*A*

_{1},

*B*

_{1}, and

*A*

_{2}, can be determined from the impermeable condition at the surface and coupling conditions between the two layers.

*z = −D*),

*z =*0), mass continuity requires

*j*, that is,

*m*

_{0}, the wave amplitude maximizes. For

*N*

_{1}

*= N*

_{2}

*= N*and in the absence of background winds, Eq. (7) reduces to

*k*

_{0}, the characteristic horizontal wavenumber, is defined as

According to Eq. (8), the wavenumber of a resonant mode is complex; that is, *x* as ~*e*-folding decay distances for mode 1, *L _{d} =* 1/

*k*

_{1}, are approximately 5

*L*

_{0}, 15

*L*

_{0}, and 232

*L*

_{0}, respectively, where

*n*= 0, 1, 2, … and

*n*). Apparently it requires

*D*and buoyancy frequency

*N*

_{1}and is independent of

*N*

_{2}.

The impact of friction on the timing and characteristic time scales of diurnal waves induced by land–sea differential heating has been discussed by R83 and Dalu and Pielke (1989). In this study, the inclusion of Rayleigh friction (i.e.,

### b. Trapped waves in FFT solutions

*x*= 0 with a coastal transition zone (CTZ) between 0 and −

*a*. A fast Fourier transform is used to calculate

*q*(

*x*) slowly decreases toward zero so that periodic boundary conditions can be applied along the lateral walls. The half domain size

*L*must be large enough (i.e.,

*L*≫

*a*) to minimize the impact of lateral boundaries on waves and circulations near the CTZ. For this study, there are 2

^{14}grid points in the

*x*direction with a grid spacing Δ

*x*= 0.05

*a*in this section and Δ

*x*= 0.1

*a*in section 4, yielding

The governing parameters of this two-dimensional two-layer system include *U _{j}*,

*N*, and

_{j}*α*, where index

_{j}*j*=1, 2 for the lower and upper layers, as well as the boundary layer depth

*D*, inversion strength

*g*′, Coriolis parameter

*f*, and horizontal dimension of the coastal transition zone

*a*. For simplicity, we use

*Q*

_{0}= 1.2 × 10

^{−5}W m

^{−3},

*D*= 1 km, and

*a*= 60 km for solutions presented in the rest of this section. The sensitivity of diurnal waves to

*a*in a uniform atmosphere has been discussed by R83 and Qian et al. (2009). For this study, the CTZ width has been varied from 10 to 100 km, and we find that, while the wave amplitude (i.e., maximum

*w*) tends to decrease with increasing

*a*, the dominant wave mode and general wave patterns are not sensitive to

*a*. The other control parameters are grouped into the following nondimensional parameters: Froude numbers Fr

*/*

_{j}= U_{j}*N*, Coriolis parameter

_{j}DWe first examine the impact of a thin inversion layer on the characteristics of the land–sea breeze by contrasting three solutions, corresponding to 1) no inversion, 2) a weak inversion (*g*′ = 0.05 m s^{−2}, *g*′ = 0.2 m s^{−2}, *U _{i}* = 0,

*N*= 0.01 s

_{i}^{−1},

^{−1}and are confined within 300 km of the CTZ (Figs. 2a,b). It is evident that a low-level inversion tends to trap diurnal waves (Figs. 2c–f). Even with a weak inversion, a leaky trapped wave forms in the boundary layer with phase lines nearly vertically oriented, and the wave amplitude decays with offshore distance. Above the inversion perturbations leaking through the inversion propagate up with an elevation angle of

*k*can be estimated from Fig. 3 by assuming an exponentially decay,

_{i}*e*-folding decay distance

*L*, estimated from linear model solutions over a range of inversion strengths, are in agreement with those of mode 1 obtained by directly solving Eq. (8); for example, for

_{d}*L*estimated from the FFT solution is ~2000 km, comparable to ~15

_{d}*L*

_{0}given by Eq. (8). For a moderate inversion, power spectra of the perturbation energy from the FFT results indicate two energy maxima corresponding to wave modes

*n*= 0 and 1. The shorter wave (i.e.,

*n*= 1) maximum is significantly more pronounced, as its wavenumber is closer to the characteristic wavenumber,

Cross sections of *w* (grayscale, increment 10 mm s^{−1}) and *u* (contours with negative values dashed, interval 2 m s^{−1}) valid at local noon time for *w* at 800 m (grayscale, contour interval 10 mm s^{−1}) and *u* at the surface (contours with negatives dashed, interval 3 m s^{−1}). The white contours correspond to *w* = 5 (solid) and −5 (dashed) mm s^{−1}.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

Cross sections of *w* (grayscale, increment 10 mm s^{−1}) and *u* (contours with negative values dashed, interval 2 m s^{−1}) valid at local noon time for *w* at 800 m (grayscale, contour interval 10 mm s^{−1}) and *u* at the surface (contours with negatives dashed, interval 3 m s^{−1}). The white contours correspond to *w* = 5 (solid) and −5 (dashed) mm s^{−1}.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

Cross sections of *w* (grayscale, increment 10 mm s^{−1}) and *u* (contours with negative values dashed, interval 2 m s^{−1}) valid at local noon time for *w* at 800 m (grayscale, contour interval 10 mm s^{−1}) and *u* at the surface (contours with negatives dashed, interval 3 m s^{−1}). The white contours correspond to *w* = 5 (solid) and −5 (dashed) mm s^{−1}.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

The 800-m vertical velocity *w* (mm s^{−1}) is plotted vs cross-shore distance valid at local noon for (a) four solutions corresponding to ^{−1} (i.e., *N*_{1} = *N*_{2} = *N* = 0.006 and 0.012 s^{−1} for the first pair and *N*_{1} = *N*_{2} = 0.01 s^{−1}, *D* = 500 and 2000 m for the second pair.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

The 800-m vertical velocity *w* (mm s^{−1}) is plotted vs cross-shore distance valid at local noon for (a) four solutions corresponding to ^{−1} (i.e., *N*_{1} = *N*_{2} = *N* = 0.006 and 0.012 s^{−1} for the first pair and *N*_{1} = *N*_{2} = 0.01 s^{−1}, *D* = 500 and 2000 m for the second pair.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

The 800-m vertical velocity *w* (mm s^{−1}) is plotted vs cross-shore distance valid at local noon for (a) four solutions corresponding to ^{−1} (i.e., *N*_{1} = *N*_{2} = *N* = 0.006 and 0.012 s^{−1} for the first pair and *N*_{1} = *N*_{2} = 0.01 s^{−1}, *D* = 500 and 2000 m for the second pair.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

The dependence of resonant wave modes on *N* and *D* can be inferred from Fig. 1. In general, for a given inversion strength, trapped waves are shorter for a less stable or a shallower boundary layer and accordingly propagate at a slower speed. The variations of *k _{i}* with

*N*and

*D*are more complicated. The characteristic wavenumber

*k*

_{0}is inversely proportional to

*N*and

*D*. On the other hand, a smaller

*N*or

*D*implies a larger

*N*= 0.012 s

^{−1}decays faster with distance than

*N*= 0.006 s

^{−1}, likely because

*N*(i.e., ~1/

*N*

^{2}) than

*D*(i.e., ~1/

*D*).

Equatorward of 30° latitude, the wavelength of the trapped diurnal wave mode for a given inversion is larger at a higher latitude. The wavelength for

As in Fig. 2, but for

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

As in Fig. 2, but for

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

As in Fig. 2, but for

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

To examine the impact of the vertical variation of stability on wave characteristics, two solutions are shown in Fig. 5 corresponding to the buoyancy frequency pairs 0.006/0.012 and 0.006/0.025 s^{−1}. As predicted by condition (7), even in the absence of an inversion, a more stable layer aloft tends to partially trap diurnal waves. For a given *N*_{1}, the wave trapping is more efficient with a larger *N*_{2} and accordingly the wave amplitude decays offshore more slowly (Figs. 5c,d). On the contrary, if the upper layer is less stable, no clear trapped waves are present (not shown). The characteristics of the trapped waves from four solutions are further compared in Fig. 6. According to Eq. (7), for waves trapped by a more stable layer aloft, the wavelength of the trapped wave is proportional to *N*_{1} and independent of *N*_{2}. In addition, a BL top inversion seems to be more effective in trapping diurnal waves; waves decay much slower under a weak inversion with *g*′ = 0.05 m s^{−2} than under a stable layer with buoyancy frequencies typical for the tropical and subtropical lower troposphere (i.e., *N*_{1} = 0.01 s^{−1}, *N*_{2} = 0.012 s^{−1}). Further inspection indicates that in the presence of a moderate inversion, the characteristics of the trapped wave are relatively insensitive to the stability above the inversion.

As in Fig. 2, but for *N*_{1} = 0.006 s^{−1}, and *N*_{2} = (a),(b) 0.012 s^{−1} and (c),(d) 0.025 s^{−1}. The *w* increment is 20 mm s^{−1}and the white contours correspond to *w* = 10 (solid) and −10 mm s^{−1} (dashed).

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

As in Fig. 2, but for *N*_{1} = 0.006 s^{−1}, and *N*_{2} = (a),(b) 0.012 s^{−1} and (c),(d) 0.025 s^{−1}. The *w* increment is 20 mm s^{−1}and the white contours correspond to *w* = 10 (solid) and −10 mm s^{−1} (dashed).

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

As in Fig. 2, but for *N*_{1} = 0.006 s^{−1}, and *N*_{2} = (a),(b) 0.012 s^{−1} and (c),(d) 0.025 s^{−1}. The *w* increment is 20 mm s^{−1}and the white contours correspond to *w* = 10 (solid) and −10 mm s^{−1} (dashed).

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

The 800-m vertical velocity (mm s^{−1}) valid at local noon is plotted vs offshore distance for four solutions corresponding to the buoyancy frequency pair *N*_{1}/*N*_{2} = 0.003/0.01, 0.006/0.01, and 0.006/0.012 s^{−1}. The corresponding curve from *N*_{1}/*N*_{2} = 0.006/0.012 s^{−1} and ^{−2} (i.e.,

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

The 800-m vertical velocity (mm s^{−1}) valid at local noon is plotted vs offshore distance for four solutions corresponding to the buoyancy frequency pair *N*_{1}/*N*_{2} = 0.003/0.01, 0.006/0.01, and 0.006/0.012 s^{−1}. The corresponding curve from *N*_{1}/*N*_{2} = 0.006/0.012 s^{−1} and ^{−2} (i.e.,

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

The 800-m vertical velocity (mm s^{−1}) valid at local noon is plotted vs offshore distance for four solutions corresponding to the buoyancy frequency pair *N*_{1}/*N*_{2} = 0.003/0.01, 0.006/0.01, and 0.006/0.012 s^{−1}. The corresponding curve from *N*_{1}/*N*_{2} = 0.006/0.012 s^{−1} and ^{−2} (i.e.,

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

Finally, LSB solutions can be significantly modified by friction in the BL. The wave solutions shown in Figs. 2–6 are nearly inviscid (i.e.,

As in Figs. 2e and 2f, but for *w* increment is 10 mm s^{−1} and the white contours correspond to 5 (solid) and −5 mm s^{−1} (dashed). The *u* contour intervals are (a),(c) 2 and (b),(d) 3 m s^{−1}, respectively.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

As in Figs. 2e and 2f, but for *w* increment is 10 mm s^{−1} and the white contours correspond to 5 (solid) and −5 mm s^{−1} (dashed). The *u* contour intervals are (a),(c) 2 and (b),(d) 3 m s^{−1}, respectively.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

As in Figs. 2e and 2f, but for *w* increment is 10 mm s^{−1} and the white contours correspond to 5 (solid) and −5 mm s^{−1} (dashed). The *u* contour intervals are (a),(c) 2 and (b),(d) 3 m s^{−1}, respectively.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

The 800-m vertical velocity (mm s^{−1}) valid at local noon is plotted vs offshore distance for three solutions corresponding to

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

The 800-m vertical velocity (mm s^{−1}) valid at local noon is plotted vs offshore distance for three solutions corresponding to

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

The 800-m vertical velocity (mm s^{−1}) valid at local noon is plotted vs offshore distance for three solutions corresponding to

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

### c. Background winds

The characteristics of the trapped waves appear to be fairly sensitive to the background wind speed. We start by examining two examples corresponding to *g*′ = 0.2 m s^{−2} and an off-shore wind speed of 4 or 8 m s^{−1}. The nondimensional Rayleigh friction coefficient is 0.1 in the lower layer and 0.01 in the upper layer, mimicking a frictional BL and the free atmosphere aloft. For both cases, trapped waves are evident downwind of the CTZ with much weaker waves above the inversion (Fig. 9). When the background wind is weak, there are two distinctive trapped modes, a short one and a long one (Figs. 9a,b), both of which decay downstream. The long wave has a wavelength substantially longer than the corresponding trapped mode with zero background winds, and decays slower with offshore distance. The estimated propagation speed (Fig. 9b) is 7.2 m s^{−1}, comparable to the Doppler-shifted wave speed, *U* = 8 m s^{−1}, the upstream propagating short wave mode dominates and the long wave is indistinguishable (Figs. 9c,d). For

(left) Vertical cross sections of *w* (grayscale, increment 10 mm s^{−1}) and *u* (contour interval 2 m s^{−1}, negative dashed) valid at local noon are shown for *U* = (a) 4 and (c) 8 m s^{−1}. The solid and dashed white contours correspond to *w* = −5 and 5 mm s^{−1}. (right) The corresponding Hovmöller diagrams of *u* at the surface (contour interval 3 m s^{−1}) and *w* at the 800-m level.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

(left) Vertical cross sections of *w* (grayscale, increment 10 mm s^{−1}) and *u* (contour interval 2 m s^{−1}, negative dashed) valid at local noon are shown for *U* = (a) 4 and (c) 8 m s^{−1}. The solid and dashed white contours correspond to *w* = −5 and 5 mm s^{−1}. (right) The corresponding Hovmöller diagrams of *u* at the surface (contour interval 3 m s^{−1}) and *w* at the 800-m level.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

(left) Vertical cross sections of *w* (grayscale, increment 10 mm s^{−1}) and *u* (contour interval 2 m s^{−1}, negative dashed) valid at local noon are shown for *U* = (a) 4 and (c) 8 m s^{−1}. The solid and dashed white contours correspond to *w* = −5 and 5 mm s^{−1}. (right) The corresponding Hovmöller diagrams of *u* at the surface (contour interval 3 m s^{−1}) and *w* at the 800-m level.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

As in Fig. 9, but for *w* (grayscale) and *u* (contour) increments in (c) and (d) are reduced to half of those in (a) and (b).

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

As in Fig. 9, but for *w* (grayscale) and *u* (contour) increments in (c) and (d) are reduced to half of those in (a) and (b).

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

As in Fig. 9, but for *w* (grayscale) and *u* (contour) increments in (c) and (d) are reduced to half of those in (a) and (b).

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

Similarly, for a two-layer atmosphere with a less stable layer below (no inversion), the trapped wave modes are modulated by the steady background wind as well (Fig. 11). For both weak and strong background wind cases, the wavelengths of the trapped waves are longer than the corresponding no background wind solutions. Compared to solutions with a moderate inversion (Fig. 9), waves in the more stable upper layer are considerably stronger.

As in Figs. 9 and 10, but for *N*_{1}/*N*_{2} = 0.006/0.012 s^{−1}.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

As in Figs. 9 and 10, but for *N*_{1}/*N*_{2} = 0.006/0.012 s^{−1}.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

As in Figs. 9 and 10, but for *N*_{1}/*N*_{2} = 0.006/0.012 s^{−1}.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

### d. Further remarks on diurnal waves trapped in BL

In summary, we have demonstrated that diurnal waves can be partially trapped within a BL by a sharp inversion or simply a more stable layer aloft, and the characteristics of trapped wave modes are dependent on the stratification, BL depth, inversion strength, large-scale winds, the earth’s rotation, and BL friction. The diurnal waves in the FFT solutions are consistent with the resonant wave condition (7).

The results above suggest that a sharp inversion, in general, is more efficient than a more stable upper layer in trapping wave energy. On the other hand, when the inversion depth is thin, the stability in the BL increases accordingly. This occurs in nature typically associated with BL top stratus or stratocumulus, which tends to keep the inversion thin through radiative cooling. For example, the research flight 6 during VOCALS sampled a potential temperature jump of more than 12 K across a depth of 50 m (Wood et al. 2011).

Nevertheless, while using an infinitesimally thin potential temperature jump to represent a BL top inversion greatly simplifies the problem, questions arise regarding the sensitivity of a trapped wave to the inversion depth and, accordingly, the applicability of the theory. One way to answer these questions is to formulate a three-layer model similar to the one in appendix B except that the heating source remains in the lowest layer. [Note that the solution coefficients with heating residing in the BL are different than those in Eqs. (B6)–(B10)]. The second layer represents an inversion of a finite depth *H*. We keep the potential temperature change across the second layer *H* is small enough, solutions converge toward the corresponding infinitesimally thin inversion solution. Despite substantial differences between the two three-layer models, the resonant condition is the same [i.e., Eq. (B11) in appendix B]. It is instructive to examine the behavior of the resonant condition (B11) in the small *H* limit. For a fixed potential temperature change across the second layer a leftward-propagating, we have

It is noteworthy that trapped wave modes are independent of the vertical heating profile. The half-sine heating profile used in the current study is largely chosen for the sake of simplicity. For example, if replacing the half-sine heating profile with a quarter-cosine one. that is, *m*_{0} = *π*/2*D*, in the BL, the resulted solutions are qualitatively similar to the corresponding ones with a half-sine heating profile. It is interesting to contrast the trapped diurnal waves examined in this study with classical stationary trapped waves in the lee of mountains (Scorer 1949). Trapped mountain waves owe their existence to a wave ducting layer aloft usually characterized by a smaller Scorer parameter, *U* or smaller buoyancy frequency *N* in the ducting layer), and the trapped wave becomes evanescent in the ducting layer where the Scorer parameter is smaller than the horizontal wavenumber. The ducting layer for the nonstationary diurnal waves is an inversion or a more stable layer aloft, and waves are only partially trapped associated with partial wave reflection at the interface.

## 4. Waves induced by a deep heating source

Now we consider a three-layer atmosphere, composed of a shallow lower layer, a middle layer where diurnal heating is located, and a deep upper layer. The objectives are to examine the characteristics of diurnal waves excited by a deep mixed layer in the lower to middle troposphere often observed over tropical deserts or high topography, and to understand the frictional BL effect on the offshore propagation of diurnal waves, especially the offshore decay distance.

*N*,

_{j}*U*, and

_{j}*α*, where the layer index

_{j}*j*= 1, 2, and 3. Assuming a simple half-sine heating profile,

*z*= 0 and

*z*=

*H*, the solutions to Eq. (2) are

Again, the radiation condition has been applied to the third layer. The five constant coefficients (i.e., *A _{j}* and

*B*,

_{j}*j =*1, 2, 3) can be obtained by letting

*z = −D*and applying the mass and pressure continuity conditions (appendix A) to the lower and upper interfaces. Solutions in the physical space are evaluated using inverse fast Fourier transforms with those coefficients in appendix B.

The horizontal distribution of the heating function and domain configuration are identical to those in the previous section except for Δ*x* = 0.1*a*. The other fixed parameters include *Q*_{0} = 1.2 × 10^{−5} W m^{−3}, *H* = 5 km, which is comparable to the depth of the upward motion layer documented by Garreaud and Muñoz (2004). In the remainder of this section, a number of solutions are presented with emphasis on the impact of the frictional boundary layer, vertical stratification variation, and steady background winds on wave characteristics such as the wavelength, speed, and the *e*-folding decay distance at the surface and the boundary layer top level.

We start by comparing a solution corresponding to *N*_{1} = *N*_{2} = *N*_{3} = 0.01 s^{−1}, zero background wind, and the BL depth *D* = 0 (Fig. 12a) with the no-inversion BL heating solution in section 3 (Figs. 2a,b). The two are qualitatively similar except that the former has a much longer wavelength due to the deeper heating source and the Coriolis effect. Consequently, diurnal waves induced by the deep heating source propagate faster with a phase speed of ^{−1} (Fig. 12b) as opposed to a few meters per second for typical BL waves, and they have a significantly larger *e*-folding decay distance. As noted in R83, the characteristic horizontal length scales for these two solutions are

Cross sections of *w* (grayscale, increment 5 mm s^{−1}) and *u* (contour interval 1 m s^{−1}, negative values dashed) valid at local noon from (a) a no-BL solution (*D* = 0) and (c) a 1-km-deep viscous *w* = 2.5 (solid) and −2.5 (dashed) mm s^{−1}.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

Cross sections of *w* (grayscale, increment 5 mm s^{−1}) and *u* (contour interval 1 m s^{−1}, negative values dashed) valid at local noon from (a) a no-BL solution (*D* = 0) and (c) a 1-km-deep viscous *w* = 2.5 (solid) and −2.5 (dashed) mm s^{−1}.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

Cross sections of *w* (grayscale, increment 5 mm s^{−1}) and *u* (contour interval 1 m s^{−1}, negative values dashed) valid at local noon from (a) a no-BL solution (*D* = 0) and (c) a 1-km-deep viscous *w* = 2.5 (solid) and −2.5 (dashed) mm s^{−1}.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

The inclusion of a 500-m-deep frictional BL modifies wave patterns within the BL and does little to perturbations above the BL (Figs. 12a,c). As shown in the corresponding Hovmöller diagrams, the surface winds and BL top vertical motion are weaker in the presence of a frictional BL. However, the BL appears to have little impact on the dominant wavelengths, propagating speeds, and offshore decay distances (Figs. 12c,d). The influence of a frictional BL on offshore decay of diurnal waves is further illustrated in Fig. 13, which shows the offshore variation of the 800-m vertical velocity at noontime from four solutions. While it is evident that *w* is smaller in the presence of a deeper and more frictional BL, the offshore decay distances from the four solutions are comparable. The weak dependence of the *e*-folding decay distance on the BL characteristics is likely because the diurnal waves are largely located in the free atmosphere and only the small portion inside the BL is subjected to the BL friction.

The 800-m vertical velocity valid at local noon is plotted vs offshore distance for four solutions with different BLs.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

The 800-m vertical velocity valid at local noon is plotted vs offshore distance for four solutions with different BLs.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

The 800-m vertical velocity valid at local noon is plotted vs offshore distance for four solutions with different BLs.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

*k*is proportional to the BL friction (i.e.,

_{i}*D*/

*H*), and the horizontal wavenumber. Accordingly, for a smaller

*D*/

*H*, the variation in the

*e*-folding decay distance (i.e.,

The influence distance or *e*-folding decay distance can be substantially modified by background winds and vertical variations in stratification. In the presence of steady seaward winds, most perturbations appear in the downwind side of the source and the *e*-folding decay distance becomes significantly larger owing to the reduced elevation angle of the wave beam over sea (Fig. 14). Similar to BL diurnal waves, diurnal waves excited by a deep mixed layer can be trapped in the lower troposphere in the presence of a more stable layer aloft (Fig. 15). The stability in the shallow BL and the uppermost layers has little impact on the trapped wave mode. Instead, the wavenumber of the trapped wave is determined by the depth and stratification in the lower troposphere; that is,

As in Fig. 12, but for *U* = (a),(b) 3 and (c),(d) 8 m s^{−1}. The BL is 500 m deep with *w* (grayscale) are (a),(c) 5 and (b),(d) 2 mm s^{−1}. The white contours correspond to (b),(d) ±1 and (a),(c) ±2.5 mm s^{−1}.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

As in Fig. 12, but for *U* = (a),(b) 3 and (c),(d) 8 m s^{−1}. The BL is 500 m deep with *w* (grayscale) are (a),(c) 5 and (b),(d) 2 mm s^{−1}. The white contours correspond to (b),(d) ±1 and (a),(c) ±2.5 mm s^{−1}.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

As in Fig. 12, but for *U* = (a),(b) 3 and (c),(d) 8 m s^{−1}. The BL is 500 m deep with *w* (grayscale) are (a),(c) 5 and (b),(d) 2 mm s^{−1}. The white contours correspond to (b),(d) ±1 and (a),(c) ±2.5 mm s^{−1}.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

As in Fig. 14, but for *N*_{1} = *N*_{2} = 0.006 s^{−1} and *N*_{3} = 0.012 s^{−1}. The background wind *U* = (a),(b) 0 and (c),(d) 3 m s^{−1}. The grayscale (i.e., *w*) increments are (a),(c) 10 and (b),(d) 5 mm s^{−1}, respectively.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

As in Fig. 14, but for *N*_{1} = *N*_{2} = 0.006 s^{−1} and *N*_{3} = 0.012 s^{−1}. The background wind *U* = (a),(b) 0 and (c),(d) 3 m s^{−1}. The grayscale (i.e., *w*) increments are (a),(c) 10 and (b),(d) 5 mm s^{−1}, respectively.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

As in Fig. 14, but for *N*_{1} = *N*_{2} = 0.006 s^{−1} and *N*_{3} = 0.012 s^{−1}. The background wind *U* = (a),(b) 0 and (c),(d) 3 m s^{−1}. The grayscale (i.e., *w*) increments are (a),(c) 10 and (b),(d) 5 mm s^{−1}, respectively.

Citation: Journal of the Atmospheric Sciences 69, 5; 10.1175/JAS-D-11-0220.1

## 5. Discussion and conclusions

A linear theory of diurnal waves in a multiple-layer atmosphere is developed to investigate offshore propagating waves induced by land–sea differential heating. Dynamics and characteristics of waves induced by two types of heating sources are examined: a two-layer atmosphere with a diurnal heating source confined within the shallow lower layer and a three-layer atmosphere with a heating source located in the relatively deep middle layer. The former aims to advance the understanding of the flow response to a boundary layer heating, typical in subtropical or midlatitude coastal areas, and the focus is on the impact of BL top inversion, vertical stratification variation, and large-scale winds on waves partially trapped inside the BL. The latter investigates diurnal waves induced by a deep heating source located above a boundary layer, mimicking a deep well-mixed layer extending up to midtroposphere over a tropical continent or over tropical or subtropical high terrain, with emphasis on the stratification variation and BL friction effects on offshore decay distance of surface winds and BL top vertical motion.

It has been demonstrated that, with a heating source confined in the atmospheric BL, the excited diurnal waves can be partially trapped by a BL top inversion or a more stable layer above. The trapped wave, characterized by vertically oriented phase lines, propagates away from the wave source, and the wave amplitude decays exponentially with offshore distance in accordance with the leaking of wave energy into the free atmosphere above. The resonant diurnal wave modes are sensitive to the inversion strength, BL depth, and the stratification in the BL. In general, under a stronger inversion, the wave amplitude decays slower with downstream distance as a smaller fraction of wave energy leaks through the inversion. It is noteworthy that, for a given inversion, the trapping condition may admit more than one solution, and the dominant wave mode has a wavelength comparable to the characteristic horizontal wavelength, which is dictated by the ratio of the buoyancy and diurnal frequencies and the dimensions of the heating source. The relative importance of an inversion is measured by the dimensionless inversion strength,

The wavelength and the wave speed of the resonant wave increase with latitude up to 30°, beyond which diurnal waves become evanescent. It is also evident that the trapped wave mode decays with offshore distance more slowly in a subtropical area than over a tropical area, given other conditions being the same. While a moderate BL top inversion extends perturbations (i.e., in terms of surface winds and vertical undulation of the inversion) far offshore through partial trapping, BL friction could substantially shorten the *e*-folding decay distance.

The dominant resonant wave mode can be dramatically modified by steady background winds. When the background winds are weak, there are two dominant wave modes: the shorter wave tends to propagate upstream and decays with offshore distance faster and the longer wave propagates downstream and decays much more slowly. When the background winds are strong, the short-wave mode becomes dominant. Nevertheless, in the presence of steady background winds, perturbations are enhanced in the downwind side of the heating source. This is consistent with previous observations, which showed that diurnal perturbations over tropical oceans are in general more pronounced to the west of continents, associated with prevailing trade winds. Our solutions also suggest that, in the presence of steady offshore winds, perturbations can extend far offshore, associated with partially trapped diurnal waves over high-latitude areas (i.e., polarward of 30° latitude). However, as demonstrated by Dalu and Pielke (1989), it may take days for diurnal waves to become established far offshore.

Compared to diurnal waves trapped in a BL, waves excited by a 5-km-deep heating source located above the BL have much longer wavelengths, propagate faster, and decay with offshore distance more slowly. As an example, for *N* = 0.01 s^{−1} and *H* = 5 km, the characteristic wavelength is ^{−1} at the equator to 22 m s^{−1} at 20° latitude, again comparable to observations and numerical simulations. The *e*-folding decay distance is *O*(1000 km) in the absence of a frictional BL and, as demonstrated in section 4, a shallow frictional BL plays a relatively insignificant role in shortening the offshore decay distance as only a small portion of waves is within BL and subjected to BL dissipation. The *e*-folding decay distance is generally longer in the presence of steady seaward winds. It has also been demonstrated that a more stable upper troposphere tends to partially trap diurnal waves and therefore increase the *e*-folding decay distance. The nearly vertically oriented upward motion segment of a trapped wave in the lower to middle troposphere resembles the observed and simulated “upsidence waves” as well (Garreaud and Muñoz 2004; Wood et al. 2009).

## Acknowledgments

This research was supported National Science Foundation (ATM-0749011) and by the Office of Naval Research (ONR) program elements (PE) 0602435 N and 0601153 N. The author has greatly benefited from discussions with Drs. Shouping Wang, James Doyle, and Ronald Smith. Dr. Rene Garreaud and an anonymous reviewer provided very helpful comments.

## APPENDIX A

### Matching Conditions along an Interface

*i*located at

*z = Z*the vertical velocity

_{i}*w*is related to the vertical displacement of a material surface

_{i}*i*and

*i*+ 1 are the indices for layers below and above the interface

*i*, respectively.

## APPENDIX B

### Coefficients for the Three-Layer Solutions

*D*= 0), Eq. (B11) reduces to

*N*

_{3}

*> N*

_{2}.

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