1. Introduction
Broadly speaking, the sea-breeze circulation consists of the local- and regional-scale responses to differential surface heating between the land and sea. This circulation plays a role in many atmospheric and oceanic phenomena in coastal areas, including thunderstorm initiation, the modulation of air pollution and fog, and the driving of wind-forced ocean currents. As such, the sea breeze has been studied extensively from a number of perspectives, including observational (Fisher 1960; Davis et al. 1889; Finkele et al. 1995; Miller et al. 2003; Puygrenier et al. 2005), numerical (Pearce 1955; Pearce et al. 1956; Estoque 1961; Fisher 1961; Antonelli and Rotunno 2007; Fovell 2005; Zhang et al. 2005), analytical (Jeffreys 1922; Haurwitz 1947; Schmidt 1947; Pierson 1950; Defant 1951, 658–672; Walsh 1974; Drobinski and Dubos 2009), and laboratory studies (Simpson 1997; Cenedese et al. 2000; Hara et al. 2009). Recent reviews are given by Miller et al. (2003) and Crosman and Horel (2010).
Most of the aforementioned studies have focused on local aspects of the sea breeze, since these local phenomena have the greatest impact on coastal populations. However, a number of recent studies have also considered the sea breeze from a mesoscale wave perspective, particularly in relation to near-shore convection in the tropics. As shown by Imaoka and Spencer (2000), Yang and Slingo (2001), Mapes et al. (2003a,b), and others, convection over the tropical oceans shows a distinct diurnal signal near coastlines, with systems initiating over land during the evening hours and then propagating offshore throughout the late evening and early morning. Yang and Slingo (2001) and Mapes et al. (2003b) have suggested that this propagating convective signal is likely associated with a mesoscale wave response, as produced by the diurnal heating over land. Similar suggestions have been made for propagating signals over the continents, particularly near prominent mountain ranges (Zhang and Koch 2000; Koch et al. 2001; Carbone et al. 2002; He and Zhang 2010).
The mesoscale wave dynamics of the sea breeze has been considered by only a handful of studies, mostly in the context of linear theory (Dalu and Pielke 1989; Rotunno 1983, hereafter R83). To keep the problem tractable, these studies have been cast in terms of highly idealized models, including resting background states and simple static stability profiles. The results show that for the tropics, the linear response consists of propagating internal gravity waves, while for latitudes greater than 30° the response is trapped to the coastline. An extension of the theory to include background winds was carried out recently by Qian et al. (2010, hereafter QEZ10). However, other aspects of the linear problem have received relatively little attention, including the effects of basic-state shear, static stability variations, and near-coastal terrain. The connection between the linear theory and the fully nonlinear phenomena of the sea breeze has also remained uncertain.
The goal of the present study is to explore the effect of an inland mountain range on the mesoscale aspects of the sea-breeze response, including both linear and nonlinear model calculations. As seen in the results of Imaoka and Spencer (2000) and Yang and Slingo (2001), the regions with strongest near-coastal diurnal signals are often associated with significant inland topography. In the present study, this topography is taken to be a plateau, with length scales modeled loosely on the western Columbia case considered by Mapes et al. (2003a,b) and Warner et al. (2003). The effects of the plateau are considered sequentially, beginning with the simple linear theory of QEZ10, as modified to include the topography. Nonlinear effects are then considered through experiments in which the heating amplitude and plateau height are both varied. All experiments are carried out both with and without background winds.
As in previous wave theory studies, the sea breeze is modeled as the response to a diurnally oscillating heat source. This simplification means that the study has intermediate complexity, having ties to both the simple linear wave problems, as well as the more complicated, nonlinear phenomena found in the real world. To reduce the parameter space and keep the model experiments tractable, the study is also limited to nonrotating flows (i.e., the equatorial problem).
Details of the physical setup are described in the following section. Section 3 considers the effects of topography in the linear context, focusing on both small heating amplitudes and small topography. The transition to nonlinear dynamics and intermediate topography is considered in section 4, while section 5 considers the nonlinear dynamics for large mountain heights. The final section gives a summary and discussion.
2. Model description and scaling
a. Basic physics
The scalings for this study are based on the linear theory for the sea breeze over flat terrain, as described by R83 and QEZ10. In the linear case, the depth scale is set by the H, while the depth-to-width aspect ratio as determined by the gravity wave dispersion relation is δ = ω/N. Together these two scales determine a length scale of H/δ = NH/ω. Scales for the remaining variables are then as listed in Table 1. Further discussion of these scales can be found in QEZ10.
Using the scales from Table 1 in (1)–(3) yields five nondimensional control parameters, while the terrain definition [(5)] adds four additional parameters. Together, these nine parameters completely determine the disturbance parameter space. Descriptions for the parameters are as listed in Table 2. In the present study, only three of the nine parameters will be varied: the heating amplitude ϵ, the plateau height 
Figure 1 illustrates the geometry of the model problem for the sea breeze with (Fig. 1c) and without (Fig. 1a) the plateau, as well as a control case in which the heating distribution extends to infinity in both directions (Fig. 1b).
The diabatic heat source [contour interval (c.i.) = 0.2] for (a) the sea-breeze case with no terrain, (b) the case of a plateau with no coastal zone, and (c) the sea-breeze case with the plateau included. The plateau height is 
Citation: Journal of the Atmospheric Sciences 69, 1; 10.1175/JAS-D-11-011.1
b. Model details
The simulations are calculated using the nonhydrostatic, compressible Boussinesq model described in Epifanio and Qian (2008). Acoustic propagation is handled using the time-splitting method of Klemp and Wilhelmson (1978). Terrain is included using the terrain-following coordinate of Gal-Chen and Somerville (1975). The model includes a Smagorinsky-type eddy viscosity, based loosely on the formulation of Lilly (1962). A weak background viscosity is also applied, with Reynolds number of Re = U3 × (κN2L)−1 = 225. The lower boundary conditions are free-slip and thermally insulating, as implemented for finite-slope topography by Epifanio (2007).
The calculations are computed in a 2D domain with horizontal extent 300L (or 20Lp) and depth 18.75H. Damping layers are imposed for the outer 48L of the domain in x and the upper 8.75H in z, with a radiation condition applied at the upper boundary. The horizontal grid spacing is Δx = 0.2L. The vertical grid spacing is Δz = 0.05H at the ground, with vertical stretching factor of 1.005. Such small vertical grid spacing is needed to fully resolve the narrow disturbance ray paths for U = 0, and to resolve the flow near the ground for U ≠ 0. In dimensional terms, L = 10 km, H = 800 m, N = 0.01 s−1, Δx = 2 km, and Δz = 40 m at the ground. The terrain parameters are given by Ls = 15 km, Lp = 150 km, and Lc = 90 km.
c. Linear calculations
In addition to the nonlinear calculations, the model described above can also be run using linearized dynamics. The model can be linearized about any arbitrary background state, as described further in section 3. The model can also be run in hydrostatic mode, which is useful when comparing to analytic calculations.
The linear model was verified for the case of flat terrain (i.e., 
Comparison of the nonviscous vertical velocity (c.i. = 0.08) with background wind (a)–(c) 
Citation: Journal of the Atmospheric Sciences 69, 1; 10.1175/JAS-D-11-011.1
3. Linear wave response with small terrain
In this section, the wave disturbance produced by the heat source [(6)] is explored in the context of linearized dynamics. The computations include both background winds and coastal topography and thus form a bridge between the simple 
a. Linear calculations
The problem of flow past coastal terrain has two potential wave sources: (i) the background flow past the topography and (ii) the heating gradients due to the coastline and the terrain. Here our interest is mainly the second source. To isolate this source, we let the background state be a steady mountain wave over the plateau (as computed through a nonlinear model run) and then consider the linear disturbance produced when the heat source [(6)] is added.
With the terrain added, the linear problem can no longer be addressed analytically because of the nonconstant coefficients in (7)–(11). To overcome this problem, the solutions are computed using the linearized model described in section 2c. The computation methodology involves two steps. First, using the nonlinear model, a simulation is computed without the heat source until a steady-state mountain wave is achieved over the plateau. A second simulation is then computed in which the model is linearized about the steady state from the first model run. The oscillating heat source is added and the linear model solves (7)–(11) until a steady oscillation is achieved.
For 
Following R83 and QEZ10, we refer to the 
b. Resting background state
For 
Figures 3a–c show the linear wave response for the case 
Linear vertical velocity response to the diurnal heating (a)–(c) with the coastline and (d)–(f) without the coastline at ωt = π/2 [c.i. = 0.1 in (a)–(c) and (f); c.i. = 0.04 in (d) and (e)]; 
Citation: Journal of the Atmospheric Sciences 69, 1; 10.1175/JAS-D-11-011.1
Note that the maximum w disturbance occurs where the disturbance ray paths intersect over the plateau. At larger heating amplitudes this maximum appears to trigger convective overturning.
c. Mountain-wave background state
Figure 4 shows the linear disturbance produced by the heat source for the case 
Linear disturbance with 
Citation: Journal of the Atmospheric Sciences 69, 1; 10.1175/JAS-D-11-011.1
Figures 4a and 4b show that the effect of the terrain in this case is to introduce a wave disturbance over the two slopes of the plateau. The disturbance amplitude increases with terrain height (as expected), and for a given terrain height the disturbance is significantly larger than for 
As shown in the appendix, the three terrain effects outlined in section 3a can be unambiguously distinguished when the terrain height is small. Figure 5 shows the three effects in isolation for the case 
The three effects of topography for the case with 
Citation: Journal of the Atmospheric Sciences 69, 1; 10.1175/JAS-D-11-011.1
Finally, linear calculations with larger mountain heights suggest that the disturbance becomes unstable once the mountain height reaches some critical value (roughly for 
4. Linear-to-nonlinear transition of wave response to small topography
The calculations described above give some insight into the effects of terrain on the sea breeze as considered in the linear context. However, the real problem is strongly nonlinear. In this section we consider several examples of the nonlinear behavior of the sea-breeze wave response by computing numerical simulations with varying interior heating amplitude. The emphasis is put on the transition of the system from linear to nonlinear. To keep consistent with the linear part, the results in this part are restricted to relatively small topography.
As in the previous work, we briefly check the reliability of the nonlinear model by comparing to the linear problem when the heating amplitude is very small. The comparison between Figs. 2a–c and 2d–f shows that the nonlinear model reproduces the linear results when the small heating is applied.
a. Nonlinear dynamics for 
As a reference point, we begin by considering the 
The total vertical velocity (c.i. = 0.06; shaded region is negative) and isentropes (black solid line) for the 
Citation: Journal of the Atmospheric Sciences 69, 1; 10.1175/JAS-D-11-011.1
For small ϵ, the flow fields in Fig. 6 resemble the linear inviscid solutions of QEZ10, with differences from the linear solution due entirely to viscosity (see section 2b). As ϵ increases, the fields begin to show higher-wavenumber features (Figs. 6b,f), and as ϵ approaches ϵ ~ 1, these high-wavenumber features collapse to form fronts and associated density currents (Figs. 6c,g). During the maximum heating part of the cycle, the onshore propagating front is still in the early stages of forming, while the offshore-moving front from the previous cooling cycle has propagated well away from the coastline. The opposite is true at t = 3.5 days, when the sea-breeze front has propagated well onshore. In general, the onshore front is found to be much stronger and better defined than the leftward-moving front, and in both cases the propagation speed of the front increases with ϵ.
Figures 7a–d display the time evolution of the 
(a)–(d) Time evolution of the total vertical velocity (c.i. = 0.24; shaded region is negative) and isentropes for the case 
Citation: Journal of the Atmospheric Sciences 69, 1; 10.1175/JAS-D-11-011.1
One result of this nonlinear back-and-forth motion is that the air in the vicinity of the coastline becomes colder than it would have without the coastline [i.e., if the heat source were given by (12)]. Toward the end of the heating phase, the temperature gradient across the coastline allows the relatively colder air over the ocean to move well onshore (Figs. 7a,b). This relatively colder air then becomes even colder once the heat source switches to cooling (Figs. 7b,c). Toward the end of the cooling phase, this colder air propagates back offshore where the heating strength is much weaker, which allows the air to remain cold while the onshore air is again heated (Figs. 7c,d). The end result is a pool of colder air that propagates back and forth across the coastline.
b. Topographic effects for 
The effect of topography on the wave fields and fronts is illustrated in Fig. 8, which shows a series of simulations for increasing ϵ (top to bottom) and increasing values of 
(a) The total vertical velocity (c.i. = 0.12; shaded region is negative) and isentropes (black solid line) at t = 3.25 days with 
Citation: Journal of the Atmospheric Sciences 69, 1; 10.1175/JAS-D-11-011.1
For ϵ = 0.01 and 0.25, the effects of the mountains in Fig. 8 are similar to those seen in Figs. 3a,b (note the difference in the axis scales). However, for larger ϵ the impact of the mountains is mainly to modify the fronts. For 
Some insight into these topographic effects is provided by comparing Figs. 7e–h to Figs. 7a–d, particularly for the offshore propagating front. Shown in Figs. 7e–h is the time evolution for the 
The character of the leftward-propagating front and cold pool for the cases with and without the coastal plateau is quantitatively exhibited in Fig. 9. Shown in the figure is the position of the leftward-moving front, as measured by the most negative gradient in buoyancy, as well as the center of the oscillating cold pool, as measured by the most negative buoyancy at the ground. For the 
Time evolution of the position of the offshore propagating front (dotted line) and the center of the cold pool (solid line) with 
Citation: Journal of the Atmospheric Sciences 69, 1; 10.1175/JAS-D-11-011.1
Figures 7i–l show the time evolution for the vertical velocity field with the heated topography but without the coastline [i.e., using (12) as the heat source]. The result shows that the heated topography does not introduce the propagating fronts, as seen in the cases with the coastline included. The effects of the terrain are thus not independent of the coastline: the terrain and the coastline are both needed to produce the enhanced offshore effects.
c. Flows with background wind
To illustrate the effects of the background wind, we again consider the case with 
(a) The vertical velocity disturbance (c.i. = 0.12; shaded region is negative) and isentropes at t = 3.25 days with 
Citation: Journal of the Atmospheric Sciences 69, 1; 10.1175/JAS-D-11-011.1
The first column of Fig. 10 illustrates the case with no terrain. For small ϵ, the model solutions are well described by the linear theory of QEZ10, with the main differences again due to viscosity. As ϵ increases, an onshore-propagating front becomes apparent in the w field, even though the temperature gradient is relatively weak. Comparison with the 
The time evolution of the 
(a)–(d) Time evolution of the vertical velocity disturbance (c.i. = 0.12; shaded region is negative) and isentropes (black solid line) with 
Citation: Journal of the Atmospheric Sciences 69, 1; 10.1175/JAS-D-11-011.1
The effects of terrain on the 
As illustrated by Fig. 11, the basic dynamics of these topographic effects is similar to those seen in Fig. 8. As in the 
Figure 12 illustrates the time evolution of the offshore front, the center of the cold pool, and the most negative buoyancy in the cold pool for the case with 
Time evolution of (a) the position of the front (asterisks) and the center of the cold pool (circles), and (b) the intensity of the cold pool for the case with 
Citation: Journal of the Atmospheric Sciences 69, 1; 10.1175/JAS-D-11-011.1
Finally, Figs. 11i–l show the case with 
5. Large terrain and heating depth dependence
The previous section shows the linear and nonlinear character of the coastal flow in cases with small and moderate topography. However, in many cases, the coastal terrain is larger than the case with 
a. Resting background state
The effect of larger topography on the wave fields and fronts is illustrated for the 
Vertical velocity (c.i. = 0.12) at t = 3.25 days for the case with 
Citation: Journal of the Atmospheric Sciences 69, 1; 10.1175/JAS-D-11-011.1
For the flow offshore, Fig. 13 shows that there is a significant change in the speed of the land-breeze front between the smaller mountain case (Fig. 13a) and the case with 
In addition to the land-breeze changes at large 
b. Flows with background wind
The effect of the background wind is illustrated in Fig. 14, which shows the 
Vertical velocity (c.i. = 0.24) at t = 3.25 days for the case with 
Citation: Journal of the Atmospheric Sciences 69, 1; 10.1175/JAS-D-11-011.1
As in the 
Figure 15 illustrates the time evolution of the flow for the case with 
Vertical velocity (c.i. = 0.36) for the case with 
Citation: Journal of the Atmospheric Sciences 69, 1; 10.1175/JAS-D-11-011.1
c. Heating depth effect
The previous results are all obtained with the fixed heating depth 
Vertical velocity for h0 = 2000 m and (a),(c),(e) 
Citation: Journal of the Atmospheric Sciences 69, 1; 10.1175/JAS-D-11-011.1
With increased heating depth (at fixed Q0), the propagation speed of the disturbance is expected to increase. However, the interpretation of this increase depends on the problem. Under linear dynamics, the effect of increased H is to produce a deeper wave response, which in turn implies faster wave propagation (in a dimensional sense). However, under nonlinear dynamics the main effect is to increase the net heating (as integrated in the vertical), which in turn produces a stronger density current.
As expected, in Q0 = 8.7 × 10−6 m s−3 cases, the front for H = 1600 m propagates faster than the one with H = 800 m (Figs. 16c,e). To determine whether this increase is a linear or nonlinear effect, the front speed in the H = 1600 m case with Q0 = 4.35 × 10−6 m s−3 is also calculated (Fig. 16a). In this later experiment, the vertical wavelength is still increased with the deeper heating depth, but the net integrated heating is the same as the Q0 = 8.7 × 10−6 m s−3 case with H = 800 m. The result shows that the reduced heating amplitude leads to the slowest propagation of the three cases considered, showing that the dynamics of the disturbance follows the expected nonlinear trend. The U = 5 m s−1 case in Figs. 16b,d,f shows a similar result.
6. Conclusions
The equatorial sea breeze was modeled in terms of an oscillating heat source over the land. Of particular interest was the effect of an inland plateau, with the slope of the plateau located 75 km from the coastline.
Under a diurnal scale analysis, the dominant control parameters for the problem include the nondimensional heating amplitude ϵ = Q0 × (N2ωH)−1, the nondimensional background wind speed 
In the linear problem, the sea breeze is treated as a small, diabatically forced disturbance, with either a rest state (for 
The transition to nonlinear effects was considered by gradually increasing ϵ, starting with the standard 
When terrain is added to the 
When an onshore wind is added (i.e., for 
Increasing the terrain height further (to 
It should be kept in mind that as with the previous linear studies, the current study is based on a number of simplifications and idealizations, including the use of a diurnally oscillating interior heat source. In the real problem, the heating is a complex combination of surface fluxes, turbulent mixing, and radiative effects, and the end result has both higher and lower frequencies (particularly higher harmonics) and a more complex spatial structure. The heating also has prominent day/night asymmetries. The experiments have also neglected the effects of surface friction, while at higher latitudes the Coriolis force plays a role as well. Any of these effects could potentially alter the amplitude and phasing of the flow, which could in turn modify the blocking and cooling mechanisms described above. The extent to which these complications impact real-world circulations remains a topic for future study.
Finally, it is worth noting that even with the restrictions given above, the results of this study have parallels in the problem of nocturnal coastal convection. Recent studies have revealed a tendency for nocturnal and early morning convection off coastlines, particularly in areas with near-coastal terrain. The convection starts at or near the coastline in the late evening hours and propagates offshore throughout the morning. While it remains unclear what drives these convective signals, the phasing and propagation of the signals is at least qualitatively consistent with our modeled land-breeze response, including the apparent role of topography.
Acknowledgments
This research was supported under NSF Grants ATM-0242228, ATM-0618662, ATM-0904635, ATM-1114849, and NSFC ATM-41105043; EPA Grant R03-0132; and the Basic Research Fund of the Chinese Academy of Meteorological Sciences Grant 2010Y002.
APPENDIX
Diagnostic Calculations of Terrain Effect
The system (A8)–(A12) shows that at leading order, the three terrain effects mentioned in section 3a can be cleanly separated: (i) the elevated heating gradients are described by the Q(1) term in (A10), (ii) the sea-breeze flow past the terrain slopes is described by (A12), and (iii) the interaction between the sea breeze and the background mountain wave is described by the advection terms on the right in (A8)–(A10). Since the system is linear, the response to these three forcing terms can be computed independently.
A companion model run with h0 = 0 gives the zero-order fields, and subtracting the two runs leaves the first-order corrections u(1), w(1), etc. Contributions from the individual forcing terms are then computed by including only the associated terms in the solution to (7)–(10), with the other forcing terms set explicitly to zero.
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