## 1. Introduction

Near-inertial oscillations have been known as energetic features of the circulation since observations by Ekman and Helland-Hansen (1931) in the Atlantic Ocean. They are seen as clockwise (counterclockwise) rotating, near-circular horizontal currents in the Northern (Southern) Hemisphere (Webster 1968; Munk and Phillips 1968). Many observational, theoretical, and numerical studies have suggested that wind is one of the primary energy sources for the near-inertial motions (e.g., Munk and Wunsch 1998). They can be generated by sudden change of the local surface wind stress (e.g., passage of a front/storm/hurricane) or periodic wind forcing associated with the sea breeze near the coast. Locally generated near-inertial motions can propagate equatorward of the critical latitude in the form of internal waves up to thousands of kilometers along the ray path (e.g., Kroll 1975; Gill 1984; Kunze 1985; Garrett 2001; Alford 2003). The critical latitude (30° for sea breeze) for inertial internal waves is the latitude at which the wave frequency matches the local Coriolis frequency leading to a change in character of the wave solution. As the inertial energy cascades into small-scale motions or produces local vertical shear, it can dissipate and contribute to the ocean mixing.

In the coastal regions, the periodic sea breeze has been observed to drive significant near-inertial motions (Zhang et al. 2009; Rosenfeld 1988; DiMarco et al. 2000; Simpson et al. 2002; Hunter et al. 2007). A detailed analysis of sea-breeze spatial structure and associated sea-breeze-driven near-inertial motions from a 30-month in situ observation period on the Texas–Louisiana shelf in the Gulf of Mexico (GOM) is reported in Zhang et al. (2009) and motivates the present study. At 30° latitude, the diurnal period of sea-breeze forcing coincides with the Coriolis frequency of planetary rotation resulting in a resonant effect that leads to a maximum ocean near-inertial response, which has been confirmed on the Texas–Louisiana shelf (∼30°N; Zhang et al. 2009; DiMarco et al. 2000) and off the coast of Namibia (∼30°S; Simpson et al. 2002). Because these energetic near-inertial motions have first-baroclinic vertical modal structure (Zhang et al. 2009), they have the potential to promote vertical mixing through a shear instability. Observations on the Texas–Louisiana shelf in the summer months show that a ∼4 m s^{−1} sea breeze (weak in comparison to synoptic winds) can efficiently deepen the summer mixed layer (Zhang et al. 2009). This enhanced mixing by a relatively weak sea breeze near 30° latitude is a principal motivation of this study.

To date, the modeling studies of the coastal near-inertial ocean response to wind forcing have focused on one- and two- dimensional simulations. Analytical work (e.g., Orlic 1987) and single-point models (e.g., Simpson et al. 2002) suggest that the coastal ocean response to wind forcing has a first-baroclinic modal structure, with the magnitude of the currents in the lower layer comparable to that in the surface mixed layer, driven by the pressure gradient imposed by the no-flow condition at the land boundary. Cross-sectional two-dimensional models have been used to study the coastal generation and offshore propagation of internal waves and to examine the interaction between the near-inertial internal waves and alongshore flows (e.g., Tintore et al. 1995; Federiuk and Allen 1996). Chen and Xie (1997) reproduced the cross-shelf structure of observed near-inertial oscillations on the Texas–Louisiana shelf using a cross-section model forced by an impulsive wind. In a more recent work, Xing et al. (2004) found that although the major features of the synoptic wind-induced deepening of the mixed layer could be reproduced by the single-point model, the absence of propagating internal waves and associated mixing lead to a sharper mixed layer than in the three-dimensional model.

Most of the previous modeling studies on wind-driven inertial motions have focused on the ocean response to an impulsive wind. Little modeling work has been done to study the broadscale and three-dimensional near-inertial ocean response to sustained periodic sea-breeze forcing near the coast. Recent papers by Simpson et al. (2002) and Rippeth et al. (2002) use an idealized two-layer model to explain the observed vertical structure of the ocean response to the sea breeze. The goal of this paper is to explore the latitudinal dependence of lateral propagation of sea-breeze-driven near-inertial motions, and to offer an explanation for the enhanced vertical mixing by sea breeze near the critical latitude based on an examination of the energy budget. Nonlinear models ranging from a simple one-dimensional model to a three-dimensional model of the GOM, incorporating realistic topography are utilized. We will examine the results of numerical model fields including sea surface elevation, temperature, currents, stratification, shear, and vertical eddy viscosity (as quantified by vertical mixing coefficients in this paper). Comparisons with current and hydrographic observations collected during the Texas–Louisiana shelf Circulation and Transport Processes Study (LATEX; Nowlin et al. 1998; DiMarco et al. 1997) in the GOM are made but are limited to the evidence of the vertical mixing processes driven by the sea breeze near 30°N.

We will begin with a description of the numerical model in section 2. Section 3 discusses the results from a one-dimensional model to show the maximum near-inertial ocean response to the sea breeze at the critical latitude. This one-dimensional model is able to demonstrate the resonance issue but not the lateral energy flux process. Subsequently, three-dimensional idealized models are used in section 4 to illustrate the latitudinal dependence of lateral propagation of sea-breeze-driven near-inertial motions in the form of Poincare waves and to explain the enhanced vertical mixing by these sea-breeze-driven motions at the critical latitude. We present the results from the three-dimensional simulation with the realistic bathymetry of the GOM in section 5 as a realistic application near the critical latitude to show the significance of these sea-breeze-driven motions on shelf dynamics, and also to test whether the curvature of the coastline influences wave generation. Discussions are given in section 6, where the limitations of simple one-dimensional models are addressed. Conclusions are made in section 7. The appendix contains a description of how to estimate the lateral energy flux by internal Poincare waves.

## 2. Numerical model

The simulations were performed using the Regional Ocean Modeling System version 3.0 (ROMS 3.0). ROMS is a free-surface and terrain-following hydrodynamic ocean model widely used by the scientific community for a diverse range of applications (Shchepetkin and McWilliams 2005; Haidvogel et al. 2000). In this paper, one- and three-dimensional idealized simulations and three-dimensional, realistic simulations are conducted.

For the three-dimensional idealized simulations, the water depth is 200 m constant with 20 vertical layers. The vertical resolution is about 5 m in the pycnocline, and 15 m below that. The numerical model domain has 300 points in the latitudinal direction ranging from 18° to 42°N, and 160 points in the longitudinal direction, with a uniform resolution of 10 km × 10 km. The background horizontal viscosity is set to 5 m^{2} s^{−1}. The western boundary is a no-slip wall, while the other open boundaries are radiation with a sponge layer. A Flather (1976) condition with no mean barotropic background flow is used for the two-dimensional velocities and free surface. The sponge layer in the southern open boundary contains 30 grid points, with 10 grid points in the eastern and northern open boundaries. The horizontal viscosity is significantly enhanced in the sponge layers, by a factor of 100 at the exterior edge of the sponge layers, and decreases linearly to the background value at the interior boundary of the sponge layers. The common turbulence closure scheme *k*–ε (Umlauf and Burchard 2003) is used to calculate vertical mixing in all the numerical experiments presented in this paper. The background or minimum vertical mixing is 5 × 10^{−6} m^{2} s^{−1} for both momentum and tracers. These and other parameters used by the closure scheme are the default parameters for ROMS version 3.0.

For the three-dimensional realistic simulations, the numerical domain covers the entire GOM, and an orthogonal-curvilinear coordinate grid is used for calculation, with the grid elements focused in the northwestern section of the gulf where grid resolution is about 5 km. The grid resolution is relatively coarse in the southeastern section of the gulf where grid resolution is about 15 km. The realistic model configuration is identical to that used for the Texas Automated Buoy System (TABS) modeling effort, a real-time nowcast/forecast of surface currents over the Texas–Louisiana continental shelf (Hetland and Campbell 2007). (Real-time TABS model results are available online at http://seawater.tamu.edu/tglo.) The lateral boundaries in the three-dimensional realistic model simulations are set to no-slip walls. For the realistic case, there are 30 vertical layers. The vertical resolution is enhanced in the pycnocline, where the vertical resolution is about ∼2–5 m on the shelf and slope areas and ∼15–20 m below that. The vertical resolution is coarser in the deep ocean. The background horizontal viscosity and the turbulence closure scheme are the same as those used in the idealized case. The horizontal scale for the first baroclinic mode in the GOM is typically ∼30 km. Therefore, our model is able to resolve the first mode, which is dominant for the sea-breeze-driven near-inertial currents on the Texas–Louisiana shelf (Zhang et al. 2009).

All numerical experiments are forced by analytical wind stress, and sea surface heat flux (including shortwave, longwave, latent, and sensible heat fluxes). Sea breeze is simulated by imposing a clockwise-rotating 24-h periodic wind field with a maximum magnitude of 4 m s^{−1} at the coast, gradually decreasing offshore by a Gaussian taper function with a scale of ∼300 km from the boundary. The 4 m s^{−1} amplitude of the sea breeze corresponds to a surface wind stress of ∼0.01 N m^{−2}. The magnitude and extent of the sea breeze is chosen to approximately simulate strong sea-breeze situations, which have been observed in the summer months in the GOM (Zhang et al. 2009). Daily diurnal cycling of the solar heat flux is represented by the positive lobes of a clipped sinusoidal signal that reaches a midday (1200 local time) maximum of 150 W m^{−2}. The heating is in phase with the sea breeze during the day. The salinity field is constant (35 psu). In the subsequent calculations, initial temperature vertical profiles are based upon climatological observations in the GOM in the summer months, and correspond to a surface mixed layer of approximately 10 m. Below 10 m, the temperature decreases exponentially over 50 m. The model uses a full equation of state to calculate density from temperature and salinity. In all numerical experiments, the model is forced from rest.

## 3. One-dimensional calculation

Initially, one-dimensional simulations are performed to investigate the ocean response to sea-breeze forcing at different latitudes in the Northern Hemisphere. Four 20-day simulations are run, in which the Coriolis frequency corresponds to four different latitudes (20°, 25°, 30°, and 40°N; otherwise the runs are identical). The temporal evolution of the current profiles at different latitudes is shown in Fig. 1, from which we can see that the ocean response is strongest at 30°N. The magnitude of the currents approaches 70 cm s^{−1} at 30°N, while it is approximately 15 cm s^{−1} at other latitudes. The ocean response decreases as the latitude of the grid is shifted from the critical latitude. This is because the sea breeze is resonant with the inertial period of the ocean at 30°N; therefore, the wind driving has an in-phase relationship with the inertial current and continuously transfers energy into the ocean. As for the temperature profiles (not shown), the mixed layer gradually deepens with time, with no further deepening after about 5 days at 20°, 25°, and 40°N. Since the magnitude of the wind is relatively weak (4 m s^{−1}; a strong sea breeze but weak in comparison to synoptic winds), little influence of the wind is seen below 15 m. However at 30°N, this sea breeze can deepen the 15°C isotherm from around 15 m downward to 25 m. This indicates that the wind at 30°N is more efficient in mixing the heat downward than at other latitudes. The distribution of the squared Brunt–Väisälä frequency (*N* ^{2}), squared shear (*S*^{2}), gradient Richardson number (Ri), and vertical mixing coefficients at 30°N all indicate the deepening of the mixed layer with time (not shown). The depth of maximum stratification coincides with the depth of maximum shear. The Ri is smallest (<0.25) above the thermocline, which is consistent with the high mixing coefficients in the upper mixed layer. The resonantly forced ocean response decreases significantly away from the critical latitude. Figure 2 is the latitudinal response function obtained from 40 simulations using the one-dimensional model and suggests that the resonant response is relatively sharp with rapid fall off in response moving away from 30°N to 30°S.

Because the one-dimensional experiment does not include the lateral pressure gradient, we do not see the occurrence of the first baroclinic mode wave structure, hence, no horizontal propagation of Poincare waves. As a consequence, the mixed layer deepens continuously with time at 30°N, and the ocean does not reach a steady state for this experiment (Fig. 1c). The mixed layer can unrealistically reach the ocean bottom (∼50 m) if the duration of forcing is allowed to persist for a long period of time. As we will discuss, these aspects are significant limitations of the one-dimensional model. Some one-dimensional mixed layer models have attempted to parameterize downward short internal wave radiation from the mixed layer using a linear Rayleigh damping term (e.g., Pollard and Millard 1970). These, however, do not account for the latitudinal complexity of long Poincare wave radiation discussed in this paper.

## 4. Three-dimensional idealized case

### a. Wave characteristics from 3D idealized simulation

To investigate the latitudinal dependence of lateral propagation of sea-breeze-driven near-inertial motions, and its effect on the vertical mixing, a three-dimensional idealized experiment is performed in a rectangular basin for 80 days. Sea-breeze forcing is applied at the western coast and decreases offshore. The numerical output in the sponge layer is excluded from all the analyses.

*ω*is the frequency of the waves,

*f*is the

*Coriolis*frequency,

*k*and

*l*are horizontal wavenumbers for wave vector

*K*,

*c*

_{0}is the phase speed of the first baroclinic mode waves,

*g*′ is the reduced-gravity acceleration that contains information about the stratification, and

*h*is the depth of the thermocline corresponding to the first baroclinic mode. These equations illustrate that group speeds in both

*x*and

*y*directions have the same dependence on latitude. Therefore, we use the term “lateral” propagation to indicate both

*x*and

*y*components of motion in this manuscript. Figure 3 shows the sea surface signature of Poincare waves generated by sea-breeze forcing at different time steps. It illustrates the latitudinal character and decay of the wave solution from 18° to 42°N. By day 10, the wave field is fully developed. Although the forcing is only applied near the western coast, the coastal ocean response propagates in the form of Poincare waves, and the near-inertial energy is redistributed and is transferred offshore. The Poincare wave solution, however, behaves quite differently poleward and equatorward of 30°. Poleward of 30°, the diurnal frequency is less than the local inertial frequency. As a consequence, the wave solution is evanescent and the coastal ocean response energy is trapped. However, equatorward of 30°, the diurnal frequency is greater than the local inertial frequency, and real wave solutions exist. Figure 3 shows that the energy can propagate farther away from the coast at lower latitudes, indicating a slower group speed of these waves near the critical latitude. However, the curvature of crest and trough lines shows that the phase speed of these waves is larger near the critical latitude. The propagation of these near-inertial motions is consistent with the wave theory, in that Poincare waves have the characteristics that the group speed decreases with increasing latitude [Eqs. (4.2) and (4.3)] making them more efficient in transferring energy offshore in the southern portion of the simulation domain. The distance that the energy can propagate based on the analytical group speed [Eqs. (4.1)–(4.4);

*C*

_{gx}

*t*, where

*t*is equal to 10, 15, 20, and 25 days, respectively) is plotted on Fig. 3 (solid line), from which we can see the offshore extent of the wave field in the model simulation agrees well with the analytical prediction.

The latitudinal distribution of temperature and vertical mixing coefficients along a transect 100 km offshore after 30 days is shown in Fig. 4. As can be seen, the sea breeze is more efficient in deepening the mixed layer near the critical latitude. From Fig. 4a, we can see that the mixed layer is deepest near 30°N reaching 32 m. It is only about 18 m deep at other latitudes. The vertical mixing coefficients are largest near 30°N. Although the forcing has the same magnitude along this transect, the vertical mixing coefficients are on the order of 0.01 m^{2} s^{−1} near 20 m at 30°N and at least an order of magnitude larger than those at other latitudes (Fig. 4b). This phenomenon can be explained by considering the energy available for vertical mixing at different latitudes. Near 30°N, the largest amount of oceanic inertial energy is generated by sea breeze because of resonance. However, the internal Poincare waves are inefficient in transferring energy offshore because of the small group speed there. This indicates that most of the inertial energy dissipates near the coast at the critical latitude, and contributes to the enhanced vertical mixing. We will quantify this phenomenon in the following section by evaluating terms in the energy equation for internal Poincare waves.

### b. Energy budget for the 3D idealized case

An analysis of energy equation illustrates the energy flux associated with Poincare waves. We feel this energy flux is resolved (horizontally and vertically) in the present model and dominated by the horizontal flux for the following reasons. Based on our numerical simulations and previous observations on the Texas–Louisiana shelf (Zhang et al. 2009), the monotonic forcing by the sea breeze is not exciting short higher-frequency and higher baroclinic mode internal waves, which can propagate steeply. The energy flux by Poincare waves (the focus of this manuscript) is thus predominately horizontally associated with their long wavelength (e.g., Kroll 1975; Garrett 2001; Kunze 1985). For a horizontal wavelength *L _{H}* of ∼100 km (Figs. 3 and 10) and vertical wavelength

*L*of

_{v}*O*(100 m; the depth of the shelf), the ratio of vertical to horizontal group speed is

*C*

_{gz}/

*C*

_{gx}=

*α*=

*L*/

_{v}*L*≈ 0.001 ≪ 1 (see section 8.4 in Gill 1982). From our model results

_{H}*C*

_{gx}∼ 30 km day

^{−1}, this implies

*C*

_{gz}∼ 30 m day

^{−1}. The vertical energy flux should thus be much smaller than the horizontal energy flux. In our model simulations, our coarsest lateral resolution along the northern and western boundaries is ∼5–10 km in horizontal, and 5 m in vertical. This resolution should be adequate to resolve the ray path associated with these estimates of

*C*

_{gx}and

*C*

_{gy}.

*E*(=KE + APE) is the energy density, KE (=

*u*

^{2}〉 + 〈

*υ*

^{2}〉 + 〈

*w*〉

^{2})/2) is the kinetic energy density, APE (=

*N*

^{2}〈ε

^{2}〉/2) is the available potential energy density, ε(

*z*,

*t*) is the vertical displacement of an isopycnal relative to its time-mean position, ∂

*E*/∂

*t*is the time rate of increase or decrease of energy density,

**u**·

**∇**

*E*is the advection of wave energy by mean velocity field,

**F**

*(=〈*

_{E}*u*′

*p*′〉 =

**c**

*) is the energy flux carried by internal waves,*

_{g}E

*τ**·*

_{s}**u**is the energy input from the wind,

*τ**·*

_{b}**u**is the energy removal by bottom drag,

*Q*

_{−}is volume-integrated energy sinks associated with the loss to turbulent dissipation, the angle brackets (〈〉) indicate averaging over a few wave periods, ∫∫∫

_{v}represents a volume integration, and ∮

_{ls}, ∮

_{ss}, ∮

_{bs}represent area integrations over the lateral surface, sea surface, and sea bottom, respectively (Nash et al. 2005; MacKinnon and Gregg 2003; Kunze et al. 2002). The vertical energy flux by Poincare waves does not contribute to this energy budget calculation because Eq. (4.5) is vertically integrated over the whole water column. The energy input by the sea surface heat flux is not included in this equation because it is an order of magnitude smaller than the mechanical energy input from the wind.

The three-dimensional model is used to produce an 80-day time series of each term in Eq. (4.5) except for the dissipation *Q*_{−}. These 80-day model outputs are then divided into twenty 4-day-long segments. Each term in Eq. (4.5) (except for *Q*_{−}) is calculated and time averaged over this 4-day period (approximately 4 inertial wave periods). We find that the advection of wave energy (**u** · **∇***E*) and the energy removal by bottom drag (*τ** _{b}* ·

**u**) are negligible because they are at least an order of magnitude smaller compared to the energy flux carried out by internal waves (

**F**

*= 〈*

_{E}*u*′

*p*′〉 =

**c**

*) and the energy input from the wind (*

_{g}E

*τ**·*

_{s}**u**), respectively. Therefore, the energy available for vertical mixing (

*Q*

_{−}) at each latitude can be considered approximately as the residual of temporal change of the total energy for internal Poincare waves (∂

*E*/∂

*t*), the energy input from the wind (

*τ**·*

_{s}**u**), and the energy removal from the lateral flux (

**F**

*= 〈*

_{E}*u*′

*p*′〉 =

**c**

*). Since it is difficult to estimate*

_{g}E**c**

*from the model output directly, we calculate the energy flux based on 〈*

_{g}*u*′

*p*′〉. A detailed description on how we estimate energy flux by internal waves is shown in the appendix.

^{−3}, while the potential energy density is ∼0.1 J m

^{−3}. The ratio of these is ∼

*O*(20–50), which is consistent with the ratio between kinetic energy and potential energy for Poincare waves predicted by linear theory (Gill 1982):This ratio becomes large at 30°N (

*ω*=

*f*). At other latitudes away from 30°N, this model KE/APE ratio is typically on the order of 1 (Fig. 5), which is also consistent with the analytical ratio [Eq. (4.6)]. As for the temporal change of the energy field, the volume-integrated total energy increases with time during the first 15 days with the gradual deepening of the mixed layer. The total energy density reaches a maximum during days 16–20, when the mixed layer depth reaches its deepest and no longer deepens. The lateral energy flux starts to overcome the energy input from the wind, and the energy density near the coast in the forcing area decreases gradually with time after day 20 (not shown).

Figure 6 shows the temporal average (over the period days 17–20) of the energy input from the wind (Fig. 6a) and energy flux by internal Poincare waves (Fig. 6b) as a function of latitude and distance from the coast. Wind energy input is maximum (∼10^{5} W) at 30°N because of the resonance and is an order of magnitude larger than values at other latitudes although the magnitude of the wind forcing is uniform with latitude. Wind energy input is approximately symmetrical with respect to 30°N (Fig. 6a). In contrast, the lateral energy flux shows an asymmetric pattern on either side of 30°N (Fig. 6b). The maximum energy flux occurs near 30°N within 300 km offshore of the forcing area (∼10^{6} W; recall that the sea breeze extends to approximately 300 km offshore). Outside the forcing area, the lateral energy flux drops dramatically (∼10^{3} W at 500 km offshore) near 30°N, indicating a large horizontal flux gradient. In contrast, the energy flux is relatively small (∼10^{5} W) equatorward of 30°N in the forcing area; however, the lateral flux has similar magnitude within and outside the forcing area. At 500 km offshore, the energy flux peaks at ∼25°N. The distribution of the magnitude of energy flux can be explained in terms of the distribution of the group speed and energy of the oceanic response to the sea breeze. Although the group speed of Poincare waves is minimum near the critical latitude [Eqs. (4.1)–(4.4); for linear solution, the group speed is zero at 30°N], the maximum energy density still makes the offshore energy flux largest at 30°N in the forcing area. Outside the forcing domain, the flux becomes much weaker because of the weak energy density and the small group speed. This indicates that near the critical latitude, the sea breeze is very efficient in generating near-inertial energy because of resonance; however, the Poincare waves are inefficient in moving the energy offshore because of the small group speed. At lower latitudes south of 30°N, the sea-breeze-driven near-inertial motions are much weaker. However, the increased group speed makes the waves more efficient in transferring energy offshore. The direction of energy flux is generally southeastward (Fig. 6b) and consistent with the sea surface elevation distribution in Fig. 3. North of 30°N, the energy flux is small and trapped because of the absence of Poincare waves.

The temporal change of the lateral energy flux at 100 and 500 km offshore is shown in Fig. 7. The horizontal flux at 100 km is at a maximum at 30°N during the whole period of the 80-day simulation. However, little energy reaches 500 km offshore at 30°N even after 80 days (Fig. 7). Near 25°N, it takes about 20 days for the energy to propagate to 500 km offshore indicating a 25 km day^{−1} group speed of Poincare waves at this latitude. The time for the wave energy to reach the 500-km transect decreases with latitude, consistent with Fig. 3 and the larger group speed at lower latitudes. The temporal change of lateral energy flux at each latitude shown in Fig. 7 is caused by temporal change of the total energy density for the internal Poincare wave fields, which increases over the first 20 days and decreases gradually afterward.

After each term in Eq. (4.5) is analyzed, the residual energy (*Q*_{−}) is calculated in the water volume from the coast to 50, 100, 200, and 500 km offshore, respectively. The residual energy is then considered as the available vertical mixing energy. The latitudinal distribution of the residual energy averaged over the period days 17–20 is shown in Fig. 8. The residual energy is an order of magnitude larger at 30°N compared to other latitudes for these four different cases. This indicates more energy dissipates locally near the coast at 30°N, while the energy available for mixing is greatly reduced away from the critical latitude. Figure 8 is shown here as an example, and during other periods the residual energy available for mixing displays similar patterns. This energy analysis explains why the idealized three-dimensional simulation shows enhanced vertical mixing and deeper mixed layer near the critical latitude (Fig. 4).

## 5. Application for the Gulf of Mexico

A three-dimensional case incorporating the realistic bathymetry of the GOM is conducted to examine the ocean response to sea breeze in the real ocean near the critical latitude. The centers of the model grid cells used are shown in Fig. 9. Sea breeze is applied in a coastal strip extending approximately 300 km from the western and northern coasts. The magnitude of sea breeze is 4 m s^{−1} at the coast, and it decreases offshore gradually by a Gaussian taper function. Sea surface heat flux forcing and other initial conditions are identical to those used in the previous cases. To isolate the sea-breeze effect, no other forcing is applied (e.g., tides, existing mesoscale flows, or buoyancy-driven river plume structures are excluded).

Our previous numerical experiments (section 4) have confirmed that the oceanic response to sea-breeze forcing depends significantly on the latitude of the coastal ocean (Figs. 1, 2, and 3). The inertial response in the form of Poincare waves occurs in the numerical simulations but is limited to latitudes equatorward of 30°N (Fig. 3). This makes the GOM particularly suited to study this phenomena in this regard, because the northern coastline of the GOM is approximately at 30°N, and spans more than 1000 km from 97° to 84°W. Nearly all of the GOM lies to the south of 30°N. The simulations described in this section consider whether or not other factors such as coastline irregularities and bathymetry alter the Poincare wave response.

Figure 10 shows the vertical component of the relative vorticity field [(∂*υ*/∂*x*) − (∂*u*/∂*y*)] at 25 m from the GOM simulation (since the Poincare wave is largely a vorticity wave, its characteristics are clearly illustrated in vorticity plots). Red (blue) indicates anticyclonic (cyclonic) vorticity and illustrates the Poincare wave field that results after 6, 8, 10, and 12 days of sea-breeze forcing, respectively. In contrast to the previous simulations, however, the Poincare wave crest structure is less regular with wave crests emanating from the curved coastal boundary. Regions of enhanced wave variability are clearly visible in the figure near DeSoto (28°N, 87°W) and Alaminos (27°N, 95°W) Canyons and east of the East Mexico Slope (between 21° and 25°N and 96°W), where coastal and bathymetric curvature is largest. The waves have wavelengths of 100 km and phase speeds of ∼50 km day^{−1} and are able to propagate into the interior of the Gulf on short time scales (i.e., order of 10 days). Regions of wave propagation transport energy offshore preferentially in the southwest, northwest, and northeast corners of the gulf. This indicates that, in addition to latitudinal dependence, the wave response is largest in three regions in the gulf where coastal bathymetry curvature is maximal. We can estimate the theoretical phase speed of these Poincare waves based on the dispersion relationship [Eq. (4.1)]. Assuming *g*′ = 0.01 m s^{−2}, *h*′ = 10 m, *k* = *l* = 1 × 10^{−5} m^{−1}, *f* = 5.7 × 10^{−5} s^{−1}, the *x* component of wave phase speed *C*_{px} ≈ 80 km day^{−1}. The celerity estimated from the model simulation is consistent with the theoretical wave theory discussed above.

Figure 11 shows the temporal evolution of the model temperature profiles at the locations of LATEX moorings 21 (28.84°N, 94.08°W) and 22 (28.36°N, 93.96°W) on the Texas–Louisiana shelf (on map in Fig. 9), which are close to the critical latitude. These locations are chosen to illustrate how vertical mixing near 30°N is influenced by water depth. The model outputs are discussed first, and observations from these mooring locations are shown and discussed later. Model mooring 21 has a total water depth of 24 m. Initially, the mixed layer depth is about 10 m, with a uniform temperature of 25°C. Below the mixed layer depth, the temperature decreases with depth, and reaches 20° near the bottom. The mixed layer depth starts to increase with time during the first two weeks after the initiation of sea breeze. After about two weeks, the water column at model mooring 21 becomes fully mixed, and the temperature changes to approximately 22.5°C throughout the water column. A 2.5°C temperature change in the mixed layer indicates the occurrence of strong vertical mixing. The ocean response shows a first baroclinic modal structure as seen previously in the three-dimensional idealized case. However, we note that after the water column becomes fully mixed, there are residual Ekman surface and bottom boundary flows with the magnitude of ∼5 cm s^{−1} (Fig. 11b). After the water column becomes fully mixed, the vertical mixing coefficients become large throughout the water column (Fig. 11c). This model result indicates that a continuous sea breeze can significantly enhance the vertical mixing in the water column on the middle Texas–Louisiana shelf during the summer months. Based on LATEX wind observations, the sea breeze can persist for 2 weeks on the Texas–Louisiana shelf without interruptions in the summer months (Zhang et al. 2009). We have observed that the stratification in the water column at the LATEX moorings 21 and 22 is significantly suppressed during those uninterrupted sea-breeze periods (section 6 for details).

The results at the location of model mooring 22 (total water depth 55 m) display a quite different situation, as the water column never fully mixes (Figs. 11d–f). However, the mixed layer can reach to 20 m after 10 days of continuous sea-breeze forcing. Afterward, the base of mixed layer starts to level off, and oscillate around 20 m. No further deepening occurs at model mooring 22 although the sea breeze still exists in the following 10 days. Comparing results at model moorings 21 and 22 illustrate that the sea-breeze deepening of the mixed layer to the bottom is limited to the shallow-water regions (<50 m).

Figure 12 is a synoptic snapshot of the vertical mixing coefficients at 15-m depth at the end of the 20-day simulation. The amplitudes are very similar at 5 and 10 m on the northern and western shelves (not shown; recall sea breeze forcing is only applied near the northern and western coasts). However, at 15 m, the vertical mixing coefficients are much larger near the northern boundary than the western boundary. This is consistent with the three-dimensional idealized simulation (Fig. 4). As discussed in the previous section, the northern shelf of the GOM is close to 30°N, where the oceanic response is strongest and a large amount of energy dissipates locally and enhances vertical mixing because of the small group speed of Poincare waves. Farther southward away from 30°N, less near-inertial energy is available for vertical mixing as a consequence of the weaker response to the sea breeze and the larger group speed of near-inertial motions at the lower latitude.

A consequence of the high vertical mixing coefficients on the northern shelf is that the water column can attain a deeper mixed layer. Figure 13 displays the temporal evolution of current profiles at different latitudes along the 200-m isobath. The mixed layer depth is indicated in Fig. 13 by the depth of velocity reversal, which is ∼30 m near 30°N (Fig. 13a) and ∼10 m near 23°N, although curvature of coastline and local mixing hotspots may contribute to these differences (Fig. 13d). The oceanic inertial response is ∼50 cm s^{−1} and strongest near 30°N (Fig. 13a), while it is less than 20 cm s^{−1} near 23°N (Fig. 13d).

## 6. Discussion

### a. Comparison with observations

Although all of the numerical simulations presented in this paper are driven by idealized forcing (e.g., the sea breeze and the sea surface heat flux), we can make some qualitative comparisons on sea-breeze-driven near-inertial motions between the LATEX observations reported in Zhang et al. (2009) and the model output shown here.

#### 1) Oceanic response to sea-breeze forcing

Observational results during summer on the Texas–Louisiana shelf (∼30°N) indicate that the ocean response to sea breeze has a first baroclinic modal structure near the coast. The coastal near-inertial ocean response is significantly enhanced near the critical latitude. Although the maximum sea breeze on the shelf is only about 4 m s^{−1}, the current can be as strong as 60 cm s^{−1} after 10 day of continuous forcing near the critical latitude. The model results shown in this paper are approximately consistent with the magnitude and duration of observed sea-breeze-driven near-inertial currents in LATEX data taken on the Texas–Louisiana shelf (Zhang et al. 2009).

#### 2) Indication of near-inertial Poincare wave propagation in the water column

Mooring data on the Texas–Louisiana shelf indicates that when there are no synoptic wind events, the sea-breeze forcing is primarily in phase with the near-inertial currents in the surface mixed layer with high and significant correlation (>0.9; Zhang et al. 2009). This indicates that most of the near-inertial energy is locally generated, which is consistent with the relatively low group speed of the Poincare waves near the critical latitude (Fig. 3).

Although Zhang et al. (2009) did not have ocean observations south of 27°N in the Gulf, results of this paper suggest that the in-phase relationship between the wind and oceanic response is likely more complicated away from 30°N, as the near-inertial band motions contain both locally and remotely generated propagating Poincare wave signatures.

#### 3) Comparison of effects on mixing

Observations on the Texas–Louisiana shelf (∼30°N) show that, strong sea-breeze-driven near-inertial motions can significantly enhance the vertical mixing in the water column by increasing the velocity shear and suppressing the stratification in the water column (Zhang et al. 2009). Figure 14a shows the time series of near-surface (3 m) near-inertial current at LATEX mooring 22 in June 1994 (see Fig. 9 for location), with the major tidal components removed using the T_Tide Harmonic Analysis Toolbox of Pawlowicz et al. (2002). Estimated amplitudes of diurnal tidal current components at mooring 22 are around 5 cm s^{−1} (Zhang et al. 2009; DiMarco and Reid 1998). Sea-breeze forcing with the magnitude of ∼4 m s^{−1} persists continuously during the period of 5–15 June 1994. As a consequence, the near-inertial current reaches ∼60 cm s^{−1} by the end of this period. Figure 14b displays the density difference between the top (3 m) and middle (23 m) meters, as calculated from the temperature and conductivity (salinity) measurements. It shows that the mean density difference (black curve in Fig. 14b) in the water column decreases from around 6.5 kg m^{−3} at the beginning of this period to approximately 2.5 kg m^{−3} near 15 June, which indicates that the sea-breeze-driven near-inertial motions have generated significant vertical mixing in the water column. The instantaneous density difference (gray curve in Fig. 14b) reaches 0 around 15 June, showing that the depth of the mixed layer has reached at least 23 m during this sea-breeze event, as opposed to the typical mixed layer depth of a few meters at mooring 22 in June. Strong diurnal variabilities in the density difference time series (gray curve in Fig. 14b) have been associated with the effects of tidal straining of the horizontally varying density field in the Irish Sea (Simpson et al. 1990; Rippeth et al. 2001; Thorpe et al. 2008). However, here the straining is caused by the sea-breeze-driven near-inertial motions.

The sea breeze is suppressed during the period of 17–20 June. The near-inertial current is reduced and the density difference returns to 6 kg m^{−3} around 22 June 1994, indicating a reestablishment of the strong stratification in the water column (Fig. 14b). This observational evidence indicates that the ∼4 m s^{−1} sea breeze on this shelf can significantly enhance the vertical mixing in the water column during the summertime. It is consistent with our three-dimensional idealized and realistic GOM experiments, which suggest enhanced vertical mixing near 30°N (Figs. 4 and 12), and our energy budget calculation, which shows that the sea breeze is more efficient in promoting vertical mixing near 30°N (section 4b).

### b. Limitation of one-dimensional model

There are several significant differences between the one- and three-dimensional simulations of oceanic response to sea breeze. First, the vertical structure of the current profiles in the three-dimensional simulation shows a first baroclinic modal structure instead of just a surface trapped jet (cf. Figs. 1 and 11). This is consistent with previous research on sea-breeze-driven near-inertial motions (e.g., Simpson et al. 2002; Rippeth et al. 2002), which has shown that the flow in the bottom layer is driven by the barotropic pressure gradient set up by the no-flow conditions at the walls. In the one-dimensional case, a pressure gradient (i.e., surface tilt or isopycnal tilt) does not exist. Therefore, the current is trapped to the surface layer where the water is directly forced by the winds. Second, the magnitude of near-inertial currents at the critical latitude in the one-dimensional simulation can reach 70 cm s^{−1}, which is much larger than that in the three-dimensional simulation (cf. Figs. 1 and 11), resulting in too much mixing. Third, the mixed layer depth reaches a steady state near the critical latitude after 10-day forcing in the three-dimensional realistic simulation (Fig. 11). However, this steady state was not established in our one-dimensional simulations (Fig. 1c).

Through a comparison between the one- and three- dimensional models, we found that the one-dimensional model has limited applicability to study the oceanic response to sea breeze near the critical latitude primarily because it cannot reproduce the propagating wave field shown in the three-dimensional model. The energy redistribution by the wave field is crucial and cannot be neglected near the critical latitude. At other latitudes, this may play a less important role because the strength of these waves is much weaker.

### c. Significance of enhanced vertical mixing by diurnal wind forcing near the critical latitude—Comparison with other critical latitude mechanisms

One of the primary findings of this manuscript is that sea breeze can significantly promote the vertical mixing near the critical latitude. Vertical mixing affects many biogeochemical, as well as physical, processes. Therefore, the seasonality of sea-breeze-induced mixing may enhance or limit these processes especially during the summer months. For example, hypoxia occurs in the summer months every year on the northern shelf of the gulf (Wiseman et al. 1997; Hetland and DiMarco 2008). This summertime sea-breeze forcing could be a mechanism to ventilate the near-bottom hypoxic water, and mitigate the negative effects of hypoxia. The role of the sea breeze in hypoxia has not yet been examined.

Mechanisms that produce enhanced localized vertical mixing have received attention recently as attempts to determine global energy dissipation locations (Munk and Wunsch 1998). However, these mechanisms have failed to account for global values needed to close the energy budget. Parametric subharmonic instability (PSI) is a recently rediscovered mechanism that can produce enhanced near-inertial baroclinic mixing near 30° latitude and is produced by the interaction of barotropic tides with steep topography (van Haren 2005, 2007; MacKinnon and Winters 2005). It is interesting to note that sea-breeze-driven near-inertial baroclinic mixing produces a comparable result for a different reason near 30° latitude. Our results indicate that in coastal regions where PSI is not expected to be a significant factor because of shallow topography and weak tides (Zhang et al. 2009), sea-breeze-driven ocean resonance may still produce enhanced vertical mixing near 30° latitude.

## 7. Conclusions and future directions

This paper illustrates the latitudinal dependence of lateral propagation of sea-breeze-driven near-inertial motions, and explains the enhanced vertical mixing by these sea-breeze-driven motions at the critical latitude using a range of nonlinear models from a simple one-dimensional model to a three-dimensional model of GOM. Figure 15 gives a schematic summary of the mechanisms and oceanic responses discussed in this paper. The results indicate that the oceanic response to the sea breeze is maximum at the critical latitude. The coastal-generated near-inertial energy is trapped to the coast poleward of 30° latitude; however, equatorward of 30° latitude, it can propagate offshore in form of Poincare waves. The GOM is particularly well suited to investigate this phenomenon owing to its proximity to 30°N. The Poincare waves at 30° latitude have the characteristics of near-zero group speed, making them inefficient in transferring the maximum oceanic energy offshore. The lateral (both *x* and *y*; for simplicity, we have shown only the *x* component of energy flux in Fig. 15, but these arguments apply equally to the *y* component as well.) energy flux convergence plus the energy input from the wind is maximum near the critical latitude, and therefore more near-inertial energy dissipates near the coast and promotes vertical mixing near 30° latitude, consistent with previous observations on the northern shelf of the GOM. Equatorward of 30° latitude, the oceanic inertial response to the sea breeze becomes much weaker. The coastal-generated energy (albeit less) is efficiently moved offshore because of the increased group speed of Poincare waves. Therefore, the local dissipation is greatly reduced.

The propagation of Poincare waves is crucial for the understanding of physics of sea-breeze-driven near-inertial motions near the critical latitude. This effect cannot be neglected, and is a significant limitation of one-dimensional simulation of the sea breeze near the critical latitude.

*ζ*denotes vorticity (Kunze 1985). In this case, the latitude resonant with sea-breeze forcing could be shifted north or south. Therefore, the resonant latitudes may not be limited to near 30° latitude as reported in this manuscript. This aspect may also be important, and warrants future investigation.

## Acknowledgments

This study is funded by the U.S. Minerals Management Service under Contracts 1435-01-05-CT-39051 and 1435-0001-30509 and by NOAA/CSCOR Award NGOMEX 2006 NA06NOS4780198. The authors thank Dr. J. A. MacKinnon (Scripps) for useful discussions on this research.

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## APPENDIX

### Procedure to Estimate the Energy Flux by Internal Poincare Waves

*p*′ and velocity

*u*′ can be inferred from density

*ρ*and velocity

*u*profiles from the model output (Nash et al. 2005). First, the density anomaly

*ρ*′(

*z*,

*t*) is calculated using the vertical displacement of an isopycnal ε(

*z*,

*t*) relative to its time-mean position asThe pressure anomaly

*p*′(

*z*,

*t*) is calculated from the density anomaly using the hydrostatic equation:The surface pressure

*p*

_{surf}(

*t*) can be inferred from the baroclinicity condition that the depth-averaged pressure perturbation must vanish:The perturbation velocity is defined aswhere

*u*(

*z*,

*t*) is the instantaneous velocity,

*u*

*z*,

*t*) is the time mean of that velocity, and

*u*

_{0}(

*t*) is determined by requiring baroclinicity:The energy flux by internal waves is then estimated as

**F**

*= 〈*

_{E}*u*′

*p*′〉 =

**c**

*, where the angle brackets (〈〉) indicate averaging over a few wave periods.*

_{g}E