## 1. Introduction

The land–sea breeze (LSB) driven by cross-shore pressure gradients associated with diurnal variation in land–sea differential heating (Atkinson 1981) has been the subject of numerous observational and numerical studies because of its important impact on weather, climate, wind energy potential, and air quality in coastal areas or around islands [for reviews, see Abbs and Physick (1992), Miller et al. (2003), and Crosman and Horel (2010)] and biological processes in near-shore waters through driving upwelling (Woodson et al. 2007). Introductory textbooks use simple schematic models to illustrate the gross features of the LSB. In reality, the LSB dynamics over complex coastlines are rather complicated on account of multiscale processes, latitudinal dependence, nonlinearity, complex coastline geometry and coastal topography, and interaction with synoptic-scale flows (Azorin-Molina and Chen 2009; Gahmberg et al. 2009). Besides observational and numerical studies, our understanding of LSBs has also been significantly advanced by analytical studies (e.g., Haurwitz 1947; Geisler and Bretherton 1969; Walsh 1974; Rotunno 1983, hereafter R83; Dalu and Pielke 1989; Qian et al. 2009, hereafter Q09). Among many such studies, the latitudinal dependence of LSB and gravity wave features were investigated by R83, who examined linear response of a uniform atmosphere at rest to a specified two-dimensional diurnal heating source. R83 demonstrated that, equatorward of the 30° parallels, the LSB response to diurnal heating takes the form of inertia–gravity waves and, for higher latitudes, the LSB is characterized by localized circulations. Recently, Q09 extended the R83 solutions to include a uniform background wind and a new family of gravity waves that owed its existence to background winds was analyzed.

For the sake of simplicity, most previous analytical or idealized modeling studies of LSBs focused on two-dimensional LSBs associated with a straight coastline. However, actual coastlines are often characterized by concave and convex shapes such as bays and peninsulas, which are often heavily populated. Case or climatological studies have shown that local LSBs, to a large degree, are controlled by coastline shapes (Banta et al. 1993; Lebassi et al. 2009). In addition, LSBs around isolated islands or elongated peninsulas are clearly three-dimensional. Some features of three-dimensional LSBs have been noted in case studies using observational data or numerical simulations (e.g., McPherson 1970; Xian and Pielke 1991; Carbone et al. 2000; Crook 2001; Robinson et al. 2011). Nevertheless, our understanding of LSBs along complex coastline is still fairly limited because of a lack of systematic studies, especially analytical studies.

The objective of this study is to shed some light on characteristics of three-dimensional LSBs induced by complex coastlines. First, the linear solutions in R83 and Q09 are extended to include the background winds, the earth’s rotation, and linear viscosity effects. Then the linear theory is applied to three-dimensional LSB perturbations induced by a sinusoidal coastline and an isolated island over a wide range of geometric parameters and at different latitudinal locations. Particularly, we are interested in the surface winds and low-level vertical velocity; the former serves as an index for near-surface LSB circulations and the latter is often related to LSB-induced convection and precipitation. To focus on three-dimensional effect, the background winds and stratification of the atmosphere are assumed to be uniform in the vertical. The complexity introduced by inversion and vertical stratification variation and wind shear is addressed in a separate paper (Jiang 2012).

The remainder of this paper is organized as the follows. The formulations of the linear theory, control parameters, general solutions in the wavenumber space, and a few two-dimensional solutions in the physical space are illustrated in section 2. Characteristics of land–sea breezes associated with a sinusoidal coastline and an isolated island over a wide range of geometric parameters are investigated in sections 3 and 4. The results are summarized in section 5.

## 2. Linear theory

### a. Linear equations and control parameters

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

*U*and

*V*are the background wind components in the

*x*and

*y*directions;

*u*, υ, and

*w*are the perturbation velocity components;

*f*is the Coriolis parameter;

*α*is the Rayleigh friction coefficient;

*q*(

*x*,

*y*) denotes normalized horizontal diurnal heating distribution,

*Z*(

*z*) describes the vertical heating profile,

^{−3}), is the maximum heating rate.

*k*and

*l*are the horizontal wavenumbers in the

*x*and

*y*directions, respectively, and

*H*

_{0}(i.e.,

*A*and

*B*can be obtained from the bottom boundary condition,

*B*= 0 (i.e., no downgoing waves; e.g., Holton et al. 2002; Q09). After some manipulation, we obtain

It is evident that the linear system (1)–(5) is governed by four frequency constants, namely the diurnal frequency *ω*, Coriolis frequency *f*, buoyancy frequency *N*, Rayleigh friction coefficient *α*; two length scales (i.e., the horizontal and vertical dimensions of the heating source, *a* and *H*_{0}); and the background wind *U* (or *V*). Five nondimensional parameters can be constructed using these seven parameters. One set of such nondimensional parameters is *Q*_{0} = 1.2 × 10^{−5} W m^{−3}, *H*_{0} = 1 km, and *N* = 0.01 s^{−1}, unless specified otherwise.

The Rayleigh damping terms in Eq. (1) provide a crude representation of atmospheric dissipation effect, which tends to reduce momentum or buoyancy anomalies. It is noteworthy that the insertion of the simple linear friction terms into Eqs. (1) yields a number of benefits. The inclusion of friction regulates solution (4) by effectively removing singularities in the inviscid limit corresponding to

### b. Straight coastline examples

*x*= 0 with a coastal transition zone (CTZ) of width

*a*. Over land,

*q*(

*x*) is unity near the CTZ and slowly decreases to zero toward the eastern domain boundary to facilitate the use of periodic lateral boundary conditions. The half domain length

*L*should be much larger than the CTZ width to minimize the lateral boundary effect. In a two-dimensional (i.e.,

_{d}*x–z*) plane, we have

*l*= 0 and

*K*

^{2}=

*k*

^{2}in Eq. (4), solutions in the physical space are obtained by applying inverse fast Fourier transform to Eq. (4) with 2

^{14}grid points in the

*x*direction and a horizontal spacing of 0.02

*a*, which yields

*L*≈ 81

_{d}*a*. Solutions in the vicinity of the coastline are found to be insensitive to the increase of the domain size, implying that the periodic boundary conditions and the slow decrease of

*q*(

*x*) away from CTZ have little impact on the LSB solutions of interest.

To illustrate the characteristics of waves and their dependence on governing parameters, vertical cross sections of horizontal and vertical velocities derived from a group of four solutions for *a* = 30 km (i.e., *x* = *a*/2, where *f* | < *ω*) and specified diurnal heating was discussed in R83. The wave phase speed *c _{p}* estimated from the slope of the major axis of the vertical motion ellipsis in a Hovmöller diagram (Fig. 2a), is about 1.7 m s

^{−1}, which is comparable to the phase speed of a wave with a vertical wavelength

*H*

_{0},

*w*maximum leads the surface wind maximum by approximately 6 h (Fig. 2a). This can be seen from the incompressible two-dimensional continuity equation,

*k**is the characteristic horizontal wavenumber. The vertical velocity at a level

*h*close to the surface is approximately given by

*u*(0) leads

*w*(

*h*) by one-quarter of a diurnal period.

Hovmöller diagram of 500-m *w* (grayscale, interval = 10 mm s^{−1}) and *u* at the sea level [contours, interval = (a),(b) 1 and (c),(d) 2 m s^{−1} with negatives dashed] corresponding to the four solutions shown in Fig. 1. The white contours correspond to *w* = 5 (solid) and −5 mm s^{−1} (dashed). The thick dashed lines in (a) and (c) indicate the axes of the wave envelopes used for the estimation of phase speed.

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

Hovmöller diagram of 500-m *w* (grayscale, interval = 10 mm s^{−1}) and *u* at the sea level [contours, interval = (a),(b) 1 and (c),(d) 2 m s^{−1} with negatives dashed] corresponding to the four solutions shown in Fig. 1. The white contours correspond to *w* = 5 (solid) and −5 mm s^{−1} (dashed). The thick dashed lines in (a) and (c) indicate the axes of the wave envelopes used for the estimation of phase speed.

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

Hovmöller diagram of 500-m *w* (grayscale, interval = 10 mm s^{−1}) and *u* at the sea level [contours, interval = (a),(b) 1 and (c),(d) 2 m s^{−1} with negatives dashed] corresponding to the four solutions shown in Fig. 1. The white contours correspond to *w* = 5 (solid) and −5 mm s^{−1} (dashed). The thick dashed lines in (a) and (c) indicate the axes of the wave envelopes used for the estimation of phase speed.

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

With *U* = −3 m s^{−1} (i.e., *k _{c}* is the critical wavenumber for downwind waves given by

Plan views of the perturbation sea level wind vectors and 500-m *w* (grayscale, interval = 15 mm s^{−1}) for the sinusoidal coastline baseline solution valid at (a) 1200, (b) 1500, (c) 1800, and (d) 2100 LST. A subdomain of 400 km × 256 km is shown. The coastal transition zones are located between the thick curves. The white contours correspond to *w* = −10 (dashed) and 10 mm s^{−1} (solid). Three points along the coastline are labeled in (a), namely the bay apex A, the tip of the peninsula T, and the midbay point M, for convenience of discussion.

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

Plan views of the perturbation sea level wind vectors and 500-m *w* (grayscale, interval = 15 mm s^{−1}) for the sinusoidal coastline baseline solution valid at (a) 1200, (b) 1500, (c) 1800, and (d) 2100 LST. A subdomain of 400 km × 256 km is shown. The coastal transition zones are located between the thick curves. The white contours correspond to *w* = −10 (dashed) and 10 mm s^{−1} (solid). Three points along the coastline are labeled in (a), namely the bay apex A, the tip of the peninsula T, and the midbay point M, for convenience of discussion.

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

Plan views of the perturbation sea level wind vectors and 500-m *w* (grayscale, interval = 15 mm s^{−1}) for the sinusoidal coastline baseline solution valid at (a) 1200, (b) 1500, (c) 1800, and (d) 2100 LST. A subdomain of 400 km × 256 km is shown. The coastal transition zones are located between the thick curves. The white contours correspond to *w* = −10 (dashed) and 10 mm s^{−1} (solid). Three points along the coastline are labeled in (a), namely the bay apex A, the tip of the peninsula T, and the midbay point M, for convenience of discussion.

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

Equatorward of the 30° latitude parallels, the inclusion of *f* only modifies wave patterns quantitatively (Fig. 1c). Offshore waves with horizontal wavenumbers in the range of ^{−1} (longer waves), respectively. Poleward of 30°, with zero background winds, *γ ^{2}* becomes negative. Consequently there is no wave solution and diurnal signals decay exponentially in the vertical (R83). In the presence of background winds, the vertical wavenumber squared remains positive for two wave families with wavenumbers in the ranges of

*k*< 0 (

*k*> 0). It is evident that the dominant signal in this solution is associated with the long waves, which propagate with a mean phase speed of about 6.2 m s

^{−1}(or 9.2 m s

^{−1}relative to the flow; Fig. 2d).

As expected, LSB perturbations can be significantly weakened by friction. When increasing

## 3. Land–sea breezes along a sinusoidal coastline

Equation (7) defines a sinusoidal coastline with a meridional wavelength 2*L _{y}*, zonal amplitude

*a*(Fig. 3a). For convenience of description, we refer to the seaward concave and convex sections as bays and peninsulas, respectively. In addition to the control parameters discussed in section 2, two coastline geometric parameters are relevant to this problem: the nondimensional bay (peninsula) width,

*β*. Solutions are obtained for a range of parameters (Table 1) by inversing Eqs. (4) using FFT over a horizontal domain of 4096 × 256 grid points with a grid spacing of

*L*/128. Again, far away from the CTZ, the heating rate over land slowly decreases to zero toward the eastern boundary of the domain so that lateral periodic boundary conditions can be applied.

_{y}The maxima of *u*, *υ*, and *w* obtained from a set of solutions for a range of coastline geometry parameters normalized by the maxima derived from the corresponding two-dimensional solution. The solutions include the baseline solution (marked with an asterisk) and other solutions with different *L _{y}* and

*β*values.

### a. Characteristics of the baseline solution for sinusoidal coastline

We start by examining characteristics of the baseline solution, corresponding to *U* = *V* = 0, *f* = 0, *L _{y}* =

*L*

_{0}= 128 km (i.e.,

*β*= 0.5, and

*a*= 30 km (i.e.,

Around sunset, the vertical velocity magnitude exhibits a maximum at about 0.5 km, and the rapid decrease of *w* aloft along the left wave beam (i.e., over the bay) is partially due to friction (Fig. 4). Along a vertical cross section near the midpoint of the southern side of the bay, the amplitude of the vertical velocity is approximately symmetric relative to the center of the CTZ (Fig. 4b), similar to a two-dimensional solution. Across the bay apex, the vertical motion over the bay is significantly stronger than that over land, associated with the seaward concave curvature of the coastline. The low-level flow is characterized by strong sea breezes below 1 km and much weaker compensative return flow aloft (Fig. 4b).

(a),(b) Vertical cross sections of *w* (grayscale, interval = 10 mm s^{−1}) and *u* (contour interval = 2 m s^{−1}; negative values are dashed) oriented east–west across (a) the bay apex A and (b) the midbay point M (see Fig. 3a for locations) valid at 1800 LST. (c),(d) The corresponding Hovmöller diagrams of 500-m *w* (interval = 15 mm s^{−1}) and surface *u*. The white contours correspond to *w* = −10 (dashed) and 10 mm s^{−1} (solid). The location of the coastline is indicated by a triangle.

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

(a),(b) Vertical cross sections of *w* (grayscale, interval = 10 mm s^{−1}) and *u* (contour interval = 2 m s^{−1}; negative values are dashed) oriented east–west across (a) the bay apex A and (b) the midbay point M (see Fig. 3a for locations) valid at 1800 LST. (c),(d) The corresponding Hovmöller diagrams of 500-m *w* (interval = 15 mm s^{−1}) and surface *u*. The white contours correspond to *w* = −10 (dashed) and 10 mm s^{−1} (solid). The location of the coastline is indicated by a triangle.

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

(a),(b) Vertical cross sections of *w* (grayscale, interval = 10 mm s^{−1}) and *u* (contour interval = 2 m s^{−1}; negative values are dashed) oriented east–west across (a) the bay apex A and (b) the midbay point M (see Fig. 3a for locations) valid at 1800 LST. (c),(d) The corresponding Hovmöller diagrams of 500-m *w* (interval = 15 mm s^{−1}) and surface *u*. The white contours correspond to *w* = −10 (dashed) and 10 mm s^{−1} (solid). The location of the coastline is indicated by a triangle.

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

Diurnal variations of vertical motion and surface winds are more evident in Figs. 4c and 4d, which show that, over the bay, the sea breeze leads the 500-m vertical velocity by approximately 6 h in phase as in two-dimensional solutions. Along a meridional section oriented across the bay apex point A or midway point M, the surface sea breeze maximum is centered in the CTZ and reaches its maximum intensity around sunset. Both the surface winds and the 500-m vertical velocity weaken while propagating away from the coastline.

### b. Latitude and coastline geometry

To examine the latitudinal dependence of the LSB over a sinusoidal coastline, we compare three solutions, namely the baseline solution (

Plan views of surface wind vectors, *w* at 500 m (grayscale, interval = 15 mm s^{−1}), and vorticity *ξ* at the sea level (contours, interval = 10^{−5} s^{−1}; negatives are dashed) for a solution with parameters identical to the sinusoidal coastline baseline solution except for *w* = 10 (solid) and −10 mm s^{−1} (dashed).

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

Plan views of surface wind vectors, *w* at 500 m (grayscale, interval = 15 mm s^{−1}), and vorticity *ξ* at the sea level (contours, interval = 10^{−5} s^{−1}; negatives are dashed) for a solution with parameters identical to the sinusoidal coastline baseline solution except for *w* = 10 (solid) and −10 mm s^{−1} (dashed).

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

Plan views of surface wind vectors, *w* at 500 m (grayscale, interval = 15 mm s^{−1}), and vorticity *ξ* at the sea level (contours, interval = 10^{−5} s^{−1}; negatives are dashed) for a solution with parameters identical to the sinusoidal coastline baseline solution except for *w* = 10 (solid) and −10 mm s^{−1} (dashed).

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

As in Fig. 5, but for *w* interval is 5 mm s^{−1} and contour interval for *ξ* is 10^{−4} s^{−1} (negatives are dashed). The solid and dashed white contours correspond to *w* = 5 and −5 mm s^{−1}, respectively.

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

As in Fig. 5, but for *w* interval is 5 mm s^{−1} and contour interval for *ξ* is 10^{−4} s^{−1} (negatives are dashed). The solid and dashed white contours correspond to *w* = 5 and −5 mm s^{−1}, respectively.

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

As in Fig. 5, but for *w* interval is 5 mm s^{−1} and contour interval for *ξ* is 10^{−4} s^{−1} (negatives are dashed). The solid and dashed white contours correspond to *w* = 5 and −5 mm s^{−1}, respectively.

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

At all three latitudes, the vertical velocity is characterized by similar ascent and descent maximum pair over the bay and peninsula (or vice versa), although the shape, amplitude, phase, and offshore propagation speed of these maxima differ. The evolution of the surface winds at the three latitudes is dramatically different. Away from the equator, positive or negative vertical vorticity maxima are present over the bay and peninsula, associated with the vertical motion maxima. Using Eqs. (4a) and (4d), when *σ* = *ω*, the vertical vorticity in the wavenumber space can be written as *l* = 0 and *σ* = *ω*, we obtain

At 48.6°N, the vertical vorticity exhibits diurnal variation similar to that at 22°N except that it leads the latter by approximately 4 h in phase. The surface winds at noon are qualitatively similar to that in late afternoon at 22°N, characterized by bay breezes perpendicular to the coastline and weak vorticity (Fig. 6). From early afternoon to evening the surface winds over the CTZ gradually turn clockwise and become nearly parallel to local coastline by 2100 LST with an anticyclonic turning over the bay and a cyclonic turning over the peninsula, qualitatively similar to that at 22°N around midnight.

The variations of the sea breeze strength, vertical motion, and phase with the latitude are further summarized in Fig. 7 based on solutions with *w* decreases from a peak at the equator by approximately 25% and 50% for *=* 1 and 2, respectively (Fig. 7). The vertical motion reaches its maximum sooner as well. The surface wind speed exhibits a maximum at

(a) The maximum *w* at 500 m, (b) the maximum sea level wind speed (dashed) and *u* component (solid), and (c) the times (LST) for the 500-m *w* (solid) and the sea level wind speed (dashed) to reach maxima are plotted over a range of normalized Coriolis parameters.

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

(a) The maximum *w* at 500 m, (b) the maximum sea level wind speed (dashed) and *u* component (solid), and (c) the times (LST) for the 500-m *w* (solid) and the sea level wind speed (dashed) to reach maxima are plotted over a range of normalized Coriolis parameters.

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

(a) The maximum *w* at 500 m, (b) the maximum sea level wind speed (dashed) and *u* component (solid), and (c) the times (LST) for the 500-m *w* (solid) and the sea level wind speed (dashed) to reach maxima are plotted over a range of normalized Coriolis parameters.

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

To demonstrate the sensitivity of the LSB and associated vertical motion to the coastline geometry, we present two sets of solutions here. In the first set, we use the same parameters as in the baseline solution except that the horizontal grid spacing varies from 0.25 to 5 km, and correspondingly, the bay width *L _{y}* increases from

*L*

_{0}/4 to 5

*L*

_{0}. The bay aspect ratio

*β*= 0.5 and the CTZ width

*a*= 30 km are fixed. In the second set, the same governing parameters as in the baseline solution are used except that

*β*varies from 0 (i.e., a straight coastline) to 1.

For a small (i.e., *υ*-wind component from the northern and southern sides of the bay have opposite signs and partially cancel out each other. Accordingly, the surface winds are oriented approximately normal to the mean coastline rather than the local coastline (Fig. 8a). In accordance with weak convergence (divergence), the ascent or descent maxima over the bay or peninsula are only slightly larger than the corresponding straight coastline solution. For an intermediate bay width (i.e., *h _{i}*, the maximum superposition occurs when the nondimensional curvature radius of the bay (peninsula),

*R*is the curvature radius. For a sinusoidal coastline defined in Eq. (7), the mean curvature radius over a segment around the bay apex can be estimated using

*y*/

*L*= 3/2) for the baseline solution is near unity. For a very large (i.e.,

_{y}Plan views of the 500-m *w* [grayscale, interval = (a)–(c) 10 and (d) 20 mm s^{−1}] and sea-level perturbation wind vectors valid at 1800 LST for (a) *L* = 0.25*L*_{0} and *β* = 0.5, (b) *L* = 5*L*_{0} and *β* = 0.5, (c) *L* = *L*_{0} and *β* = 0.25, and (d) *L* = *L*_{0} and *β* = 1. A subdomain of 400Δ*x* × 256Δ*y* is shown, where (a) Δ*x* = Δ*y* = 0.25, (b) Δ*x* = Δ*y* = 5, and (c),(d) Δ*x* = Δ*y* = 1 km. The white contours correspond to −10 (dashed) and 10 mm s^{−1} (solid), respectively.

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

Plan views of the 500-m *w* [grayscale, interval = (a)–(c) 10 and (d) 20 mm s^{−1}] and sea-level perturbation wind vectors valid at 1800 LST for (a) *L* = 0.25*L*_{0} and *β* = 0.5, (b) *L* = 5*L*_{0} and *β* = 0.5, (c) *L* = *L*_{0} and *β* = 0.25, and (d) *L* = *L*_{0} and *β* = 1. A subdomain of 400Δ*x* × 256Δ*y* is shown, where (a) Δ*x* = Δ*y* = 0.25, (b) Δ*x* = Δ*y* = 5, and (c),(d) Δ*x* = Δ*y* = 1 km. The white contours correspond to −10 (dashed) and 10 mm s^{−1} (solid), respectively.

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

Plan views of the 500-m *w* [grayscale, interval = (a)–(c) 10 and (d) 20 mm s^{−1}] and sea-level perturbation wind vectors valid at 1800 LST for (a) *L* = 0.25*L*_{0} and *β* = 0.5, (b) *L* = 5*L*_{0} and *β* = 0.5, (c) *L* = *L*_{0} and *β* = 0.25, and (d) *L* = *L*_{0} and *β* = 1. A subdomain of 400Δ*x* × 256Δ*y* is shown, where (a) Δ*x* = Δ*y* = 0.25, (b) Δ*x* = Δ*y* = 5, and (c),(d) Δ*x* = Δ*y* = 1 km. The white contours correspond to −10 (dashed) and 10 mm s^{−1} (solid), respectively.

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

The second set of solutions indicates that, regardless of the aspect ratio of the bay, LSB tends to be normal to the local coastline as long as *β* (Fig. 8; Table 1). The *w* maximum for *β* = 1 nearly triples that for a straight coastline (i.e., *β* = 0) because of the positive superposition of perturbations from the curved coastline nearby. Clearly, for the set of parameters examined, a larger inland extent induces stronger flow convergence or divergence and therefore stronger vertical motion.

### c. Background winds

To demonstrate the sensitivity of LSB circulations to steady large-scale winds and the earth’s rotation, we examine two pairs of solutions with an easterly or northerly wind for

(a),(b) Plan views of surface perturbation wind vectors, *w* at 500 m (grayscale, interval = 15 mm s^{−1}) for a solution identical to the baseline solution except for *U* = −3 m s^{−1} valid at (a) 1200 and (b) 1800 LST. (c),(d) As in (a) and (b), but for a northerly wind (i.e., *V* = −3 m s^{−1}) solution. The white contours correspond to *w* = 10 (solid) and −10 mm s^{−1} (dashed), respectively.

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

(a),(b) Plan views of surface perturbation wind vectors, *w* at 500 m (grayscale, interval = 15 mm s^{−1}) for a solution identical to the baseline solution except for *U* = −3 m s^{−1} valid at (a) 1200 and (b) 1800 LST. (c),(d) As in (a) and (b), but for a northerly wind (i.e., *V* = −3 m s^{−1}) solution. The white contours correspond to *w* = 10 (solid) and −10 mm s^{−1} (dashed), respectively.

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

(a),(b) Plan views of surface perturbation wind vectors, *w* at 500 m (grayscale, interval = 15 mm s^{−1}) for a solution identical to the baseline solution except for *U* = −3 m s^{−1} valid at (a) 1200 and (b) 1800 LST. (c),(d) As in (a) and (b), but for a northerly wind (i.e., *V* = −3 m s^{−1}) solution. The white contours correspond to *w* = 10 (solid) and −10 mm s^{−1} (dashed), respectively.

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

Plan views of 500-m *w* (grayscale, interval = 15 mm s^{−1}), *ξ* at the sea level (contours with negatives dashed) and sea-level perturbation wind vectors for *U* = −3 m s^{−1} and (c),(d) *V* = −3 m s^{−1} valid at 1200 and 1800 LST, respectively. The vorticity contour intervals are (a),(b) 1.5 × 10^{−4} and (c),(d) 3 × 10^{−4} s^{−1}. The white contours correspond to *w* = 10 (solid) and −10 mm s^{−1} (dashed), respectively.

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

Plan views of 500-m *w* (grayscale, interval = 15 mm s^{−1}), *ξ* at the sea level (contours with negatives dashed) and sea-level perturbation wind vectors for *U* = −3 m s^{−1} and (c),(d) *V* = −3 m s^{−1} valid at 1200 and 1800 LST, respectively. The vorticity contour intervals are (a),(b) 1.5 × 10^{−4} and (c),(d) 3 × 10^{−4} s^{−1}. The white contours correspond to *w* = 10 (solid) and −10 mm s^{−1} (dashed), respectively.

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

Plan views of 500-m *w* (grayscale, interval = 15 mm s^{−1}), *ξ* at the sea level (contours with negatives dashed) and sea-level perturbation wind vectors for *U* = −3 m s^{−1} and (c),(d) *V* = −3 m s^{−1} valid at 1200 and 1800 LST, respectively. The vorticity contour intervals are (a),(b) 1.5 × 10^{−4} and (c),(d) 3 × 10^{−4} s^{−1}. The white contours correspond to *w* = 10 (solid) and −10 mm s^{−1} (dashed), respectively.

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

With offshore background winds, the descent maximum over the bay is stretched seaward (i.e., downwind) and becomes weaker and more widespread (Figs. 9a,b) than in the baseline solution. On the contrary, the ascent over the peninsula becomes more localized. Compared to Fig. 3a, the surface winds over the ocean exhibit much more complicated patterns and extend far offshore, apparently associated with gravity waves aloft (Figs. 3 and 9). However, in the vicinity of the coastline, the diurnal variation of the surface winds is quite similar to that from the baseline solution. With a northerly wind, the low-level descent maximum in the bay becomes more elliptical with its major axis nearly parallel to the southern coastline of the bay (Figs. 9c,d). Correspondingly, the ascent maximum over the peninsula is now oriented along the southern coast of the peninsula. Overall, the ascent and descent patterns experience similar diurnal variations as in the corresponding zero background wind solution.

Over a subtropical area, the vertical motion and surface wind patterns in the solution with *U* = −3 m s^{−1} (Figs. 10a,b) are similar to the corresponding no wind case (i.e., Figs. 5a–c) except that perturbations are noticeably stronger offshore. For the northerly wind case (i.e., *V* = −3 m s^{−1}; Figs. 10c,d), at local noon, the surface winds in the vicinity of CTZ are characterized by a cyclonic turning over the peninsula and an anticyclonic turning over the bay. The ascent (descent) and positive (negative) vorticity maxima over the peninsula (bay) strengthen with time and closed circulations form by 1800 LST (Fig. 10d), after which both the vertical motion and vorticity maxima weaken. In early evening, a sea breeze jet appears over the apex and the southern coastline of the bay (Fig. 10d). Over the bay (peninsula) a positive (negative) vorticity maximum develops from early evening to early morning. A similar diurnal variation of vorticity occurs with

## 4. Land–sea breezes around an isolated island

*R*and a CTZ width

*a*:

*R*, CTZ width

*a*, Coriolis parameter

*f*, and the background wind speed

*U*. Solutions (4) and (5) are numerically evaluated using IFFT on a 1024 × 1024 grid domain with a grid spacing of 0.01

*R*. We first examine an island baseline solution defined by

*R*= 100 km,

*a*= 30 km,

*U*= 0, and then systemically investigate the dependence of LSB on the island size, latitude, and background winds.

### a. Characteristics of the island baseline solution

The evolution of the surface winds and 500-m vertical motion in the island baseline solution is shown in Fig. 11. Around local noon, the land breeze is ceasing and gentle lifting occurs over the island (Fig. 11a). Correspondingly, descent occurs in an annular zone offshore of the island. The sea breeze strengthens with time and reaches a maximum around sunset (Figs. 11b,c). During the same period, the upward motion over the island enhances with time and becomes progressively more localized toward the center of the island. After sunset, the sea breeze starts weakening and a ring of subsidence appears between the CTZ and the island center where ascent is still present (Fig. 11d). Around midnight, subsidence dominates over the island with a ring of gentle lifting offshore (not shown), identical to Fig. 11a except that the circulation is reversed.

(a)–(d) Plan views of surface wind vectors and 500-m *w* (grayscale, interval = 10 mm s^{−1}) valid at 1200, 1500, 1800, and 2100 LST, respectively, for the island baseline solution. Only a 300 km × 300 km subdomain is shown and the CTZ is located between the thick curves. (e) The vertical cross section of *w* (grayscale) and *u* (contours, negative values are dashed) oriented east–west through the center of the island valid at 1800 LST. (f) The distance–time plot of 500-m *w* (grayscale, interval = 20 mm s^{−1}) and *u* at surface (contours, interval = 2 m s^{−1}). The location of the island center is indicated by a triangle in (e) and (f).

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

(a)–(d) Plan views of surface wind vectors and 500-m *w* (grayscale, interval = 10 mm s^{−1}) valid at 1200, 1500, 1800, and 2100 LST, respectively, for the island baseline solution. Only a 300 km × 300 km subdomain is shown and the CTZ is located between the thick curves. (e) The vertical cross section of *w* (grayscale) and *u* (contours, negative values are dashed) oriented east–west through the center of the island valid at 1800 LST. (f) The distance–time plot of 500-m *w* (grayscale, interval = 20 mm s^{−1}) and *u* at surface (contours, interval = 2 m s^{−1}). The location of the island center is indicated by a triangle in (e) and (f).

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

(a)–(d) Plan views of surface wind vectors and 500-m *w* (grayscale, interval = 10 mm s^{−1}) valid at 1200, 1500, 1800, and 2100 LST, respectively, for the island baseline solution. Only a 300 km × 300 km subdomain is shown and the CTZ is located between the thick curves. (e) The vertical cross section of *w* (grayscale) and *u* (contours, negative values are dashed) oriented east–west through the center of the island valid at 1800 LST. (f) The distance–time plot of 500-m *w* (grayscale, interval = 20 mm s^{−1}) and *u* at surface (contours, interval = 2 m s^{−1}). The location of the island center is indicated by a triangle in (e) and (f).

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

At sunset, the most intense sea breeze occurs over the CTZ, and the depth of the sea breeze is about half of the scale height of the heating source, above which a weak return airflow is present (Fig. 11e). Wave beams emit from the CTZ and a substantially enhanced updraft maximum forms over the center of the island, due to the convergence of the landward wave beams from the circular CTZ. The diurnal variation of the LSB circulation is evident in the Hovmöller diagram (Fig. 11f). From the late morning to the late afternoon, gentle lifting occurs over land, with a maximum strengthening with time while propagating toward the center of the island. The vertical velocity in the middle of the island peaks around 1700 LST, approximately 1 h before the sea breeze reaches its maximum strength over the CTZ. The subsidence maximum propagates at a comparable speed and decays with offshore distance rapidly. It is worth noting that the horizontal dimension of LSB circulations around an island is typically smaller than an LSB over a straight coastline. Over the island, the azimuthal symmetry requires that the radial component of the horizontal wind decreases to zero at the island center, and accordingly the inland extent of a sea breeze is less than *R*. In addition to dissipation and upward tilting of wave beams, perturbations also weaken offshore due to wave energy density divergence in the horizontal. Assuming the atmosphere is inviscid, the wave energy flux across a vertical cylinder around the island should be constant for a steady wave source; that is, *r*_{0} is the radius of the cylinder and *E*(*r*, *z*) is the wave energy density. Therefore, the perturbation energy density around the island decays with the increasing offshore distance due to the horizontal divergence as 1/*r*_{0}, in addition to the decay caused by the friction and wave beam tilting as in two-dimensional solutions.

### b. Island size dependence

Four solutions have been obtained using the control parameters as in the baseline solution except with *R* = 20, 50, 300, and 500 km, respectively. In general, the diurnal variation of LSB is qualitatively similar over the range of island sizes examined. Quantitatively, the LSB intensity and associated vertical motion vary significantly with the island size. The sea breeze intensity increases with increasing island size sharply for *R* < 200 km and becomes nearly constant for larger islands (Fig. 12). Dynamically, for a small island [i.e., the nondimensional radius

The *u* maximum (m s^{−1}, solid) at the sea level and *w* maximum (mm s^{−1}, dashed) at the 500-m level are plotted vs the radius of the circular island.

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

The *u* maximum (m s^{−1}, solid) at the sea level and *w* maximum (mm s^{−1}, dashed) at the 500-m level are plotted vs the radius of the circular island.

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

The *u* maximum (m s^{−1}, solid) at the sea level and *w* maximum (mm s^{−1}, dashed) at the 500-m level are plotted vs the radius of the circular island.

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

The maximum vertical motion at the 500-m level exhibits a peak at *R* = 100 km, which decreases sharply with the increase or decrease of *R* and becomes nearly a constant for *R* > 300 km. We expect the vertical motion maximum peaks when the nondimensional island radius, *h _{i}*, which yields

*R*~ 100 km for

*h*= 500 m. For a large island (

_{i}*h*/

_{i}*H*

_{0}> 1, the vertical motion enhanced by wave beam superposition can be significantly weakened by friction.

### c. Latitude dependence

As expected, the LSB characteristics around an island are sensitive to its latitudinal location. Shown in Fig. 13 is a solution with governing parameters identical to the baseline solution except for

As in Fig. 11, but for the solution with ^{−4} s^{−1}) are superposed in (a)–(d).

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

As in Fig. 11, but for the solution with ^{−4} s^{−1}) are superposed in (a)–(d).

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

As in Fig. 11, but for the solution with ^{−4} s^{−1}) are superposed in (a)–(d).

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

The vertical cross section at 1800 LST shows that the vertical velocity reaches a maximum around 400 m, much lower than the baseline solution, and, correspondingly, the sea breeze circulation is shallower (Fig. 13e). In addition, both the surface cross-shore winds and associated vertical motion reach maxima earlier than in the baseline solution (Fig. 13f). Poleward of 30°, phases of the surface winds are dramatically different from over lower-latitude areas. Specifically, near local noon, sea breezes are well developed with little vorticity and wind direction nearly normal to the coastline (not shown). The cyclonic circulation over the island strengthens with time in the afternoon and peaks around sunset with the surface winds nearly parallel to the coastline, which is qualitatively similar to that at midnight for *<* 1.

The dependence of surface winds and vertical motion on the latitude is summarized in Fig. 14. The vertical motion maximum shows little change for

The variations of (a) the maximum vertical velocity, (b) the surface zonal wind component, and (c) the area-integrated enstrophy at the sea level (i.e., ^{2} s^{−2}) with the normalized Coriolis parameter are plotted. The local hours for the enstrophy to peak are labeled in (c) for several Coriolis parameters.

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

The variations of (a) the maximum vertical velocity, (b) the surface zonal wind component, and (c) the area-integrated enstrophy at the sea level (i.e., ^{2} s^{−2}) with the normalized Coriolis parameter are plotted. The local hours for the enstrophy to peak are labeled in (c) for several Coriolis parameters.

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

The variations of (a) the maximum vertical velocity, (b) the surface zonal wind component, and (c) the area-integrated enstrophy at the sea level (i.e., ^{2} s^{−2}) with the normalized Coriolis parameter are plotted. The local hours for the enstrophy to peak are labeled in (c) for several Coriolis parameters.

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

### d. Background winds

The diurnal variations of the surface winds and 500-m vertical velocity at the equator in the presence of a uniform background wind, *U* = −3 m s^{−1}, are shown in Fig. 15. The presence of background winds breaks the azimuthal symmetry of flow patterns. Around local noon, an arc-shaped descent zone is present along the upwind edge of the island and a much stronger ascent banner appears over the downwind side of the island. Onshore flow is evident along the CTZ, and perturbations extend far downstream of the island, associated with gravity waves (Fig. 15a). The ascent banner becomes stronger and more concentrated in early afternoon and begins to weaken after 1400 LST (Figs. 15b,c). By late evening, a narrow arc-shaped ascent zone appears over the upwind side of the island along with two ascent maxima emitting from the island flanks (Fig. 15d).

Plan views of surface wind vectors and 500-m *w* (grayscale, interval = 20 mm s^{−1}) valid at (a) 1200, (b) 1500, (c) 1800, and (d) 2100 LST for *U* = −3 m s^{−1} and

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

Plan views of surface wind vectors and 500-m *w* (grayscale, interval = 20 mm s^{−1}) valid at (a) 1200, (b) 1500, (c) 1800, and (d) 2100 LST for *U* = −3 m s^{−1} and

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

Plan views of surface wind vectors and 500-m *w* (grayscale, interval = 20 mm s^{−1}) valid at (a) 1200, (b) 1500, (c) 1800, and (d) 2100 LST for *U* = −3 m s^{−1} and

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

The earth’s rotation seems to have little impact on the diurnal variation of the ascent and descent patterns near the island except for a modest phase shift. However, the surface wind patterns are evidently more complicated because of the Coriolis force (Fig. 16). In the afternoon, a cyclonic circulation forms over the lee half of the island, strengthening with time while drifting downstream. From midnight to the next morning, the vertical motion and the mesoscale circulation reverse signs. It is noteworthy that the mesoscale circulations generated over the island can be advected far downstream.

As in Fig. 15, but for ^{−4} s^{−1}; negatives are dashed) are superposed. The white contours correspond to *w* = −10 (dashed) and 10 mm s^{−1} (solid).

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

As in Fig. 15, but for ^{−4} s^{−1}; negatives are dashed) are superposed. The white contours correspond to *w* = −10 (dashed) and 10 mm s^{−1} (solid).

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

As in Fig. 15, but for ^{−4} s^{−1}; negatives are dashed) are superposed. The white contours correspond to *w* = −10 (dashed) and 10 mm s^{−1} (solid).

Citation: Journal of the Atmospheric Sciences 69, 6; 10.1175/JAS-D-11-0137.1

## 5. Discussion and conclusions

Land–sea breezes forced by diurnal heating along a sinusoidal coastline or around an isolated island have been examined using linear theory. First, we extend the linear theory of R83 and Q09 to include steady large-scale winds, the earth’s rotation, and Rayleigh friction, all of which have significant impact on the characteristic of land–sea breezes. Then waves and circulations associated with a sinusoidal coastline and a circular island are examined over a wide range of geometric parameters.

According to our results attained from linear theory, the LSB along a sinusoidal coastline differs significantly from a straight coastline and is sensitive to the coastline geometric parameters, latitudinal locations, and the large-scale winds. Over a medium-sized bay (peninsula) defined by

Over a narrow bay (i.e.,

Local LSB circulations and offshore wave patterns can be significantly complicated by the earth’s rotation and the background winds. In general, over a higher-latitude region, the vertical motion is weaker. Along the CTZ, the surface winds are characterized by cyclonic or anticyclonic turnings over the bay (peninsula) associated with the vertical squashing or stretching of low-level flow. The vorticity over the bay or peninsula experiences diurnal variations and accordingly the surface winds rotate clockwise with time. In the presence of steady cross-coastline background winds, the LSB-related perturbations are largely confined to the downwind side of the coastline and could extend far downstream associated with inertia–gravity waves aloft. It is particularly interesting that with large-scale equatorward flow and inclusion of the earth’s rotation, closed circulations (i.e., mesoscale eddies) occur over bays or peninsulas, which is consistent with previous observations of the Santa Cruz eddy over Monterey Bay, California (Archer et al. 2005).

From early afternoon to early evening, LSB induced by a medium-sized island at the equator [i.e., *h _{i}* exhibits a peak around

The LSB circulations induced by an island are dramatically modified by the earth’s rotation. Around local noon, associated with low-level ascent, an anticyclonic eddy forms over the island with wind directions nearly parallel to the coastline. The surface winds gradually become stronger and more oriented across the coastline in the early afternoon in accordance with the decrease of vorticity over the island. In late afternoon, a transition occurs with cyclonic circulation dominant over the island and anticyclonic vorticity over the surrounding area. The presence of steady large-scale winds has a dramatic impact on low-level vertical motion. Specifically, the ascent over the island prevailing from early afternoon to early evening becomes stronger and more localized along the downwind side of the island, implying that LSB-induced or -enhanced convection and precipitation are more likely over the downwind coast of an island. In addition, in early evening, two ascent maxima are present along the flanks (relative to the prevailing winds) of the island. LSB perturbations can extend far downstream of an island associated with gravity waves. Over an island located equatorward of 30°, the cyclonic and anticyclonic circulations over the island described in the calm condition are shifted to downstream of the island. These mesoscale eddies can also be detached from the island and advected downstream.

Although only idealized coastlines are examined in this study, the linear FFT method illustrated here can be applied to real coastlines. The following parameters or data are needed for calculating a LSB driven by land–sea differential heating using this method: the mean atmospheric stratification *N*, large-scale winds (*U*, *V*), the horizontal heating distribution function *q*(*x*, *y*), the vertical scale height of the heating *H*_{0}, and the Rayleigh friction coefficient *α*. The domain should be large enough to minimize any spurious impact from the lateral boundaries. However, it should be emphasized that thermally forced circulations are often nonlinear and cannot be accurately reproduced by linear theory. For example, it has been shown in previous studies that some strong sea-breeze fronts may be better modeled as a gravity current (e.g., Sha et al. 1991), which is highly nonlinear and turbulent.

## Acknowledgments

This research was supported by the National Science Foundation (ATM-0749011) and by the Office of Naval Research (ONR) program elements 0601153 N and 0602435 N. The author has greatly benefited from discussions with Drs. Shouping Wang and James Doyle.

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