## 1. Introduction

The sea–land breeze (SLB), a common phenomenon mainly observed in the planetary boundary layer (PBL), is induced by the diurnal rhythms of the thermal contrast between land and sea. It is one of the oldest topics that meteorologists are interested in and has been studied extensively by numerous observational and numerical studies [for example, see review papers by Abbs and Physick (1992), Miller et al. (2003), and Crosman and Horel (2010)]. Moreover, analytical studies also make significant contributions for advanced understanding of the SLB. Early analytical modeling supposed that the SLB is mainly the balance between the pressure gradient caused by unequal heating and friction (Jeffreys 1922). The effect of Earth’s rotation was then taken into account in 1947. The deflecting force of Earth’s rotation (Coriolis force) was used to explain the influence of the geostrophic wind on the diurnal variations of the SLB (Haurwitz 1947) and to explain why the wind direction was not perpendicular to the coast when the SLB was blowing at full strength (Schmidt 1947). The force effect of a prescribed surface temperature function on the SLB circulation was then emphasized (Walsh 1974). The observed difference in the intensities of sea and land breezes (the sea breeze is generally stronger than the land breeze) was explained analytically by the time-varying diurnal variation of the stratification and the related variation of the eddy diffusion coefficients that were expected to give rise to a temporally asymmetrical circulation even when the surface forcing is symmetrical (Mak and Walsh 1976). The rotation of the direction of the SLB was also analytically investigated (Neumann 1977; Simpson 1996). Furthermore, the atmospheric response was discussed by the relative size of the diurnal heating and cooling frequency

For mathematical convenience, the SLB is mainly discussed in two-dimensional space—for example, in an *x*–*z* plane. The streamfunction then could be introduced to simplify equations to an equation containing only one variable. The analytical solution then could be obtained to conduct a theoretical discussion. It would be much more complex when dealing with the three-dimensional situation. Zhang et al. (1999) made a good attempt. They reduced the three-dimensional equations to a complex partial differential equation, which is so complicated that they had to solve it by a numerical relaxation method. Jiang (2012a) extended the linear solutions in Rotunno (1983) and Qian et al. (2009) to the three-dimensional SLB perturbation induced by complex coastlines. He further discussed the complexity introduced by inversion and vertical stratification variation and wind shear (Jiang 2012b). The above studies applied a Fourier transform to combine the system equations into a single wave equation that could be solved analytically under some simple circumstances. Then, characteristics of the three-dimensional SLB could be discussed. However, when applying the three-dimensional models to complex coastlines, it is difficult to obtain analytical solutions, and numerical methods have to be used.

The SLB is a mesoscale response of the atmosphere to horizontal variations in surface heating (Walsh 1974) that associates with the temperature difference between land and sea (TDLS). Therefore, it should be written as a function of the temperature distribution even though the form might be complex. For example, Haurwitz (1947) got the expressions for

## 2. Model framework

^{−1}represents the diurnal variation, Eqs. (1), (2), and (5) could be written as

To guarantee the finite integral value in Eq. (17), the vertical scale

## 3. Model results

^{−1},

^{2}s

^{−1},

^{−2},

^{−1}) blows from sea to land at the lower layer and reverses its direction with slower speed at the upper layer, accompanying strong updrafts near the island center and weak downdrafts over the sea. The

As demonstrated before,

On the other hand, the vertical variation of the SLB circulation is controlled by

## 4. WRF Model verification

In the previous section, we elegantly express the three-dimensional SLB circulation as functions of the surface temperature distribution and then discuss the basic structure of the SLB circulation. In this section, we further apply the National Center for Atmospheric Research (NCAR) Advanced Weather Research and Forecasting (WRF) Model version 3.6.1 to verify the theoretical calculation. The equation set for the WRF Model is fully compressible, Eulerian, and nonhydrostatic with a run-time hydrostatic option. The model uses a terrain-following, hydrostatic-pressure vertical coordinate with the top of the model being a constant pressure surface. The horizontal grid is the Arakawa-C grid. The time integration scheme in the model uses the third-order Runge–Kutta scheme, and the spatial discretization employs second- to sixth-order schemes. The model supports both idealized and real-data applications with various lateral boundary condition options (Skamarock et al. 2008). It has been used extensively in idealized sea breeze simulations (Antonelli and Rotunno 2007; A. J. Gibbs 2008, unpublished manuscript; Crosman and Horel 2012; Steele et al. 2013).

The WRF Model ideal sea-breeze case allows the users to gain a physical understanding of the sea-breeze itself by only considering the most basic and necessary parameters required to reproduce the appropriate features (A. J. Gibbs 2008, unpublished manuscript). Details on the WRF Model configured as a two-dimensional sea-breeze simulation are given in Table 1. The model setup is for a 2D case with 202 grid points and 35 vertical levels (*x*–*z* plane). The horizontal grid spacing is 2000 m, and the stretched vertical grid spacing is 85–130 m. The land occupies 50 grid points in the middle of the domain. The central longitude is taken as the prime meridian, which means the start hour is the local time as well as the universal time coordinated (UTC). There is a diurnal cycle, and the latitude and longitude are set for radiation to work. Full physics are employed, using the WRF double-moment (WDM) 5-class microphysics scheme, the Dudhia and Rapid Radiative Transfer Model (RRTM) radiation schemes, the revised Monin–Obukhov surface layer scheme, and the thermal diffusion land surface scheme. The ocean is initially set 7 K warmer than the land surface temperature. The simulation covers a 24-h time frame at 0000 UTC with no wind at the beginning.

WRF Model two-dimensional sea-breeze details (WRF namelist selection in parentheses).

The simulation depicts a clear picture of the diurnal evolution of the SLB circulation. A shallow weak land-breeze circulation is established over the coastline at approximately 0100 UTC. It breaks down and a sea-breeze circulation emerges simultaneously at 0900 UTC. The sea breeze strengthens and then weakens until 2100 UTC, when it is replaced by the land breeze. Two times (0300 and 1200 UTC) are selected to characterize the land-breeze and sea-breeze circulations, respectively. Figure 6 shows the near-surface temperature associated with the two cases. Because of the symmetry of land and sea distribution, the coordinate origin of the abscissa (represented by variable ^{−1}; see in Fig. 7b). The land-breeze circulation mainly dominates in the range of 20–80 km. The wind blows from land to sea at the lower level, then ascends to the higher level over the sea, then returns to the land at the higher level, descends to the lower level on the land, and finally forms a complete anticlockwise land-breeze circulation cell that could vertically extend up to about a 1000-m level. The land breeze at the lower level is stronger than the return flow at the higher level. The center of the circulation cell sits at about the 500-m level above the boundary between the land and sea.

We calculate the theoretical land breeze (see in Fig. 7a) driven by the temperature distribution in the WRF ideal simulation. It could be seen that the analytical solution is consistent with the WRF Model results qualitatively. For example, an analogical anticlockwise circulation cell horizontally ranges from about 40 to 60 km where the temperature gradient is obvious and vertically extends up to about 1000 m. The downdraft and updraft branches dominate on land and sea, respectively. The land breeze occurs at the lower level with the intensity about 0.3 m s^{−1}, and the return flow at the higher level is weak. Of course, there exists a certain discrepancy between the theoretical and the WRF Model results. For example, the analytical land breeze is slightly stronger than its WRF Model counterpart but with a narrower spatial extent.

The situation is similar in the case for the sea-breeze circulation. TDLS decreases from 8° to 0°C with increasing

Based on the comparison with the WRF Model, it could be seen that the theory is correct, and it does have the ability to characterize the SLB circulation, although it does have certain imperfectness owing to the highly simplified equations for obtaining the analytical solution.

## 5. Conclusions

The SLB circulation is an interesting phenomenon that has been extensively investigated either analytically or numerically. Recent theoretical progress has extended to the complete three-dimensional circulation in which the analytical solution could only be derived for some simple coastline cases. On the other hand, how to directly express the driving effect of the surface temperature on the SLB circulation is still an unsolved question that is worth further exploring. Therefore, we try to theoretically answer the question by applying the hydrostatic and incompressible Boussinesq equations under the stationary atmospheric background in the PBL. The time partial derivatives could be eliminated by introducing separation of variables. Then the horizontal motion could be reduced into a second-order differential equation, which could be solved analytically by just introducing a reasonable hypothesis that predicts that the surface temperature exponentially decays with height. Then the vertical motion could be solved analytically by integrating the incompressible continuity equation. Eventually, the theory elegantly exhibits the driving effect of the surface temperature on the three-dimensional SLB circulation.

According to the theory, the horizontal variations of the horizontal and vertical motions are determined by the first and second derivatives of the surface temperature distribution rather than the temperature itself. We calculate the SLB circulation in a simple circular island case (the island radius

The vertical distributions of the three-dimensional velocity components are determined by the vertical scale of the surface temperature. The

To further verify the theory, we apply the widely used WRF Model two-dimensional sea-breeze simulation to conduct an ideal experiment. An island with a size of 100 km is placed in the middle of the domain. We integrate the model for total 24 h from 0000 UTC and choose two times (0300 and 1200 UTC) to characterize the land breeze and sea breeze, respectively. At 0300 UTC, a weak land-breeze circulation is established near the boundary between the land and sea where the horizontal temperature gradient is significant. The downdrafts and updrafts occur on land and sea, respectively, with a corresponding wind blowing toward the sea at the lower level and toward land at the upper level. At 1200 UTC, a strong sea breeze with a reversed direction has replaced the weak land breeze. The colder sea air mass accompanying the sea breeze encounters the warmer land air mass to form an obvious SBF located inland. Using the near-surface temperature distribution in the WRF ideal simulation, the theory could give analogical circulation patterns corresponding to the land breeze and sea breeze, respectively. Both the horizontal scale and the intensity are consistent with the WRF Model results. Despite the fact that the theory does not predict the asymmetry of the circulation pattern and the SBF, it features the nature of the SLB circulation.

## Acknowledgments

This paper is supported by 973 Program (2014CB953903) and the Fundamental Research Funds for the Central Universities (2013YB45). The authors greatly thank Dr. Zhaoming Liang for his warm help on the WRF Model compilation and simulation. The authors appreciate the useful comments from Editor Dr. Grabowski and from two reviewers.

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