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Keywords:

  • coastal low-level jet;
  • breeze circulation;
  • linear model

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The linear model
  5. 3. Solutions for prescribed heating
  6. 4. Similarities to and differences from observations
  7. 5. Conclusion
  8. Acknowledgements
  9. References

This article investigates the linear dynamics of the sea breeze in an along-shore thermal wind shear. The present analysis shows that the sea-breeze circulation is tilted towards the slanted isentropes associated with the thermal wind. At a critical value of the thermal wind shear, the tilt of the sea-breeze circulation becomes equal to the slope of the background isentropes. The present analysis also shows a spatial shift between the heating pattern and the sea-breeze circulation. The present linear theory is then applied to interpret measurements made in the vicinity of New York City where there is a warm-season synoptic southwesterly jet. It is compared with observations and past numerical simulations. Agreement is found with respect to the enhanced along-coast wind that follows the tilted isentropes, the order of magnitude of the isentrope tilt and the clockwise rotating wind hodograph showing the jet maximum peaking at 1800 solar time. There is a disagreement between theory and observations on the phase lag between the jet maximum and the cross-shore pressure gradient maximum. However, this disagreement can reasonably be attributed to either the angle made by the synoptic jet to the coastline and/or the presence of friction. The inland spatial shift of the breeze indicated by the theory might also be indirectly confirmed by the coastal inland wind observations of a larger diurnal amplitude for a stronger synoptic jet. Copyright © 2011 Royal Meteorological Society


1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The linear model
  5. 3. Solutions for prescribed heating
  6. 4. Similarities to and differences from observations
  7. 5. Conclusion
  8. Acknowledgements
  9. References

The mesoscale structure of the sea-breeze circulation has in the past been idealized as the response of a rotating, stratified fluid to a diurnally varying differential surface heating (e.g. Walsh, 1974; Rotunno, 1983; Niino, 1987; Dalu and Pielke, 1989; Baldi et al., 2008; Qian et al., 2009; Drobinski and Dubos, 2009). These studies have been useful for the extraction of some basic qualitative features of the theoretical sea-breeze circulation, such as its aspect ratio, phase relative to the diurnal heating cycle, and modification by friction. An aspect that has received relatively less attention is the effect of the summertime land–sea temperature contrast (thermal wind) on the diurnally varying sea-breeze circulation. In this article we report on a simple extension to the theory of Rotunno (1983; R83 hereafter) to include a base-state thermal wind.

The effect of an onshore/offshore background wind on the dynamics of the sea breeze has been heavily investigated through numerical simulation (e.g. Estoque, 1962; Walsh, 1974; Pielke 1974; Bastin and Drobinski, 2006; Bastin et al., 2006) and observations (e.g. Bastin et al., 2005, 2006; Drobinski et al., 2006, 2007) and only recently through analytical techniques (Qian et al., 2009). The effect of an along-shore thermal wind has received comparatively little attention. Burk and Thompson (1996) modelled the summertime thermally balanced coastal jet on the west coast of the USA. They found that the diurnal sea-breeze circulation is superimposed on the summertime northerly coastal jet and is strongly influenced by elevated and variable coastal terrain as well as relatively cold coastal sea-surface temperatures. Recently a similar phenomenon has been identified on the east coast of the USA (although without the strong topographic effects). Colle and Novak (2010; CN10 hereafter) find that there is a warm-season southerly jet in the coastal regions of New York and New Jersey that often dislays a strong diurnal modulation. These jet winds are vertically confined to the boundary layer (< 300 m) and display a seasonal maximum occurrence in June–July suggesting that summertime land–sea temperature contrast is important to the jet occurrence. The jet events exhibit a diurnal cycle, rotating in a clockwise sense on a wind hodograph. However, compared to CN10 observations, the effect is much more obvious along the US west coast where the northerly jet and depressed marine layer is a quasi-permanent feature of the summertime flow. The coastline is straighter as well.

In the present article we present a simple theoretical description of the above-described phenomena through a modification of the theory for the two-dimensional seabreeze circulation proposed in R83. Using a simple coordinate transformation to geostrophic coordinates (Hoskins and Bretherton 1972), the solutions found in R83 can, for the simple case of spatially constant along-coast wind shear, be used directly. Upon transformation of that solution back to fixed Cartesian coordinates, the effects of constant along-coast shear on the sea breeze can be easily deduced.

In section 2 we introduce the linear model, coordinate transform and solution with its salient features described. Section 3 illustrates the solution for prescribed heating. In section 4 we compare the latter features with the modelled and observed features of the sea-breeze circulations embedded in the coastal jets of the west coast of the USA. Section 5 concludes the study.

2. The linear model

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The linear model
  5. 3. Solutions for prescribed heating
  6. 4. Similarities to and differences from observations
  7. 5. Conclusion
  8. Acknowledgements
  9. References

2.1. Equations of motion

Consider the Cartesian coordinate system shown in Figure 1 with the coastline at the origin and with the dark and light shadings representing sea and land, respectively. For the purposes of the present simple exposition, we consider the horizontally uniform, constant shear, along-coast jet

  • equation image(1)

where V0 is a constant and Λ is the vertical wind shear. Assuming thermal-wind balance we have

  • equation image(2)

where the base-state buoyancy B(x,z) ≡ gΘ/Θ0, g is the gravitational acceleration, Θ(x,z) is the base-state potential temperature and Θ0 is a constant reference value. Substituting Eq. (1) into Eq. (2) we have

  • equation image(3)

so that

  • equation image(4)

where N is the Brunt frequency (assumed constant) and B(0,0) is a constant.

thumbnail image

Figure 1. The dark and light shadings represent the cooler sea and the warmer land, respectively. The thermal wind is illustrated by the arrows indicating decreasing southerly wind with height.

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The equations of motion linearized about the base-state wind V(z) are

  • equation image(5)

where (u,v,w) are the components of the perturbation wind vector in the directions (x,y,z), respectively, ϕ = p/ρ0 is the perturbation pressure divided by a reference air density, b is the perturbation buoyancy and (Fu,Fv,Fw,Fb) are friction terms which will first be taken equal to zero. The boundary conditions are w = 0 at z = 0 and the requirement for bounded solutions as x2 + z2 [RIGHTWARDS ARROW] ∞. In the following, we consider N to be constant and prescribe Q = Q(x,z,t).

From Eq. (5), one can derive a single equation for the streamfunction ψ (u = ∂ψ/∂z and w = −∂ψ/∂x):

  • equation image(6)

Herein we take Q = H(x,z)sinωt (where ω = 2π/24 h is the diurnal frequency and t = 0 corresponds to sunrise), so that ψsinωt. Since Nequation image 10−2 s−1 and ωequation image 0.73 × 10−4 s−1, N2ω2; under the latter conditions, Eq. (6) becomes

  • equation image(7)

Following R83, we set

  • equation image(8)

where Qmax, x0, z0 are the amplitude, horizontal and vertical scales of the heating, respectively. When x0 [RIGHTWARDS ARROW] 0, Q(x,z,t) is a step function in x that decays exponentially away from the lower boundary, similar to the heating function used in Drobinski and Dubos (2009).

2.2. Non-dimensional equations

Non-dimensionalizing the governing equation (7) with

  • equation image(9)

where

  • equation image

we have

  • equation image(10)

Without the heating term, Eq. (10) is the familiar equation for symmetric instability (e.g. Bluestein, 1993, p. 317) with instability occurring if equation image. In the present application, we consider the stable case equation image and situations in which f > ω (latitude > 30°); with the latter restrictions, Eq. (10) is elliptic and may be transformed into

  • equation image(11)

where

  • equation image(12)

The coordinate transformation (12) is well known in the theory of frontal circulations, as are the solution techniques for the Poisson equation (11) (e.g. Hakim and Keyser, 2001). Here we note that Eq. (11) is identical to (14) of R83 with equation image and thus the solutions given in R83 may be used directly.

2.3. Solutions

The solution of Eq. (11) for the point source equation image far away from boundaries is the Green's function (omitting the time dependence)

  • equation image(13)

(Morse and Feshbach, 1953, p. 798). With the source located at ξ′ = ζ′ = 0, the dimensional form of the argument of the logarithm in (13) is proportional to

  • equation image(14)

where α ≡ (f2ω2)/N2 and γ ≡ Λf/N2(= (∂z/∂x)B in the light of Eq. (4)). With a rotation of coordinates, (14) can be expressed as

  • equation image(15)

where

  • equation image(16)

For typical values of the meteorological parameters, (α,γ) ≪ 1; consequently Eq. (16) indicates that ϵequation imageγ and Eq. (15) reduces to equation image. Thus we deduce that surfaces of constant ψ are ellipses, with the ratio of major to minor axes being N(f2ω2)−1/2 (as in (19) of R83), but tilt so that the major axes align with the tilt of the isentropes Λf/N2. The horizontal scale of the land/sea-breeze cell z0α1/2 is thus comparable to the Rossby radius z0N/f (Drobinski and Dubos, 2009).

The solution of Eq. (11) for the two point sources equation image, which satisfies the lower boundary condition equation image, is the modified Green's function (again omitting the time dependence)

  • equation image(17)

To help investigate the spatial dependence of the solution, we express Eq. (17) in terms of the coordinates equation image and consider a point source located at the coast equation image and at some level equation image; thus the image source is located at equation image; with the latter specifications, the right-hand side of Eq. (17) becomes

  • equation image(18)

Figure 2 displays the Green's function G0 for a source located at equation image for equation image and equation image. Several features may be deduced directly from the formulae. First, we note that G0 is symmetric with equation image and tilted for equation image, as expected from the solution for the point source discussed above. Second, we see that the horizontal decay away from the (tilted) maximum is more rapid in the case with equation image. To deduce the latter analytically, we fix the height equation image so that (18) becomes

  • equation image(19)
thumbnail image

Figure 2. Green's function (Eq. (18)) with point source located at (0,1) for (a) equation image and (b) equation image.

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The horizontal scale of G0 is therefore proportional to equation image in equation image coordinates. One can similarly examine the vertical decay for fixed equation image; setting equation image, one can show that equation image. However, when the Green's function is applied to a given heating function (see below), the vertical scale of the sea-breeze cell cannot be smaller than the vertical scale of the heating itself, and therefore the vertical scale of the sea-breeze cell will be ∼ 1, independent of equation image.

One further feature that will be useful for later interpretation is the calculation of the cross-coast velocity equation image at equation image. From Eq. (18) we deduce

  • equation image(20)

which shows that, for equation image, the maximum cross-coast wind at the ground moves inland by the distance equation image.

3. Solutions for prescribed heating

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The linear model
  5. 3. Solutions for prescribed heating
  6. 4. Similarities to and differences from observations
  7. 5. Conclusion
  8. Acknowledgements
  9. References

The solution to Eq. (11) for the heating function (Eq. (8) is given by (24) of R83, which in the present notation is

  • equation image(21)

which in equation image coordinates becomes, using Eq. (8),

  • equation image(22)

The solutions for equation image for equation image, 0.5 and 0.9 are shown in Figures 3 and 4; the zero-shear case reproduces the solution shown in figure 2a of R83. In the presence of shear, the linear response displays an asymmetric shape of the breeze cell with a seaward tilt (x > 0), which increases with the shear (∼ Λf/N2). Onshore (x < 0), the sea-breeze cell seems to be compressed to a shallower depth and to extend farther inland than in the absence of shear. However the sea-breeze aspect ratio is maintained independent of the shear, as shown previously by the scaling analysis. The intensity of the breeze also increases with shear.

thumbnail image

Figure 3. Upper, middle and lower rows display the non-dimensional streamfunction equation image, equation image and equation image wind components, respectively, at equation image. Left, centre and right columns are for equation image, 0.5 and 0.9, respectively.

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thumbnail image

Figure 4. Upper and lower rows display the non-dimensional v and b wind components, respectively, at equation image. Left, centre and right columns are for equation image, 0.5 and 0.9, respectively.

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The sea-breeze strength and direction as a function of time, visualized by hodographs, is a key aspect of the sea-breeze dynamics. Figure 5 shows the wind hodographs (polar diagram where wind direction is indicated by the angle from the centre axis and its strength by the distance from the centre), rotating clockwise from the south at equation image (sunrise), as a function of distance to the shore and for various values of the shear equation image. At a fixed height (equation image in Figure 5), the present solutions indicate that the sea breeze is weaker offshore for stronger shear and the magnitude of the diurnal wind variation decreases with increasing shear, whereas the opposite behaviour is found onshore. Indeed, onshore (x < 0), Figures 5(a) and (b) show that the magnitude of the wind fluctuations is larger in the presence of shear. Offshore, the sea breeze tilt implies that near-surface wind decreases rapidly for increasing distance from the shore. Since the tilt increases with shear, despite the intensification of the sea breeze with shear, for a given distance from the shore and a given height, the near-surface wind decreases with increasing shear. Onshore, the absence of tilt and the increase of the breeze intensity with shear implies that, for a given distance from the shore and a given height, the near-surface wind increases with increasing shear.

thumbnail image

Figure 5. Near-surface wind perturbation hodograph (equation image) starting at equation image, with data plotted every 2π/24 time units (º marker), for equation image (black), 0.5 (red) and 0.9 (green) and for (a) equation image, (b) equation image, (c) equation image and (d) equation image. In (a) and (d), the radius of the circle represents equation image, whereas in (b) and (c), it represents equation image.

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3.1. Spatial lag between heating and breeze cell

Equation (20) indicates a spatial shift between the heating pattern and the sea-breeze response. Figure 6(a) displays the horizontal wind component equation image close to the surface. It shows that the equation image disturbance is shifted onshore as equation image increases (see also Figure 3) and that the magnitude of the near-surface wind increases with increasing shear onshore while it decreases with increasing shear offshore.

thumbnail image

Figure 6. (a): Wind component equation image as a function of equation image for equation image (black), 0.5 (red) and 0.9 (green) at equation image. (b): Onshore distance equation image of the maximum wind speed equation image as a function of equation image at equation image.

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Figure 6(b) shows the position equation image of the near-surface wind maximum as a function of equation image. Consistent with Eq. (20) implying that equation image, Figure 6(b) shows evidence of a linear relationship between equation image and equation image.

4. Similarities to and differences from observations

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The linear model
  5. 3. Solutions for prescribed heating
  6. 4. Similarities to and differences from observations
  7. 5. Conclusion
  8. Acknowledgements
  9. References

In CN10's climatological study, the southwesterly near-surface jet that develops primarily during the warm season east of the northern New Jersey coast and south of Long Island ranges typically between 11 and 17 m s−1 (Figures 2 and 3 of CN10). The wind directions for the jet (Figure 6 of CN10) trace out a nearly elliptical orbit for the 24 h period, similar to the inertial rotation of the sea breeze (e.g. Neumann, 1984). CN10 discusses a geostrophic adjustment during the day for the near-surface jet, so by the end of the day the flow is quasi-balanced on a larger scale. The scale of the adjustment is relatively large (Rossby radius), so the enhanced winds extend well offshore (Figures 13d and 21a of CN10). This is also consistent with linear sea-breeze theory which shows that the horizontal scale of the land/sea-breeze cell is comparable to the Rossby radius (Drobinski and Dubos, 2009). Thus our interpretation is that the near-surface jet described in CN10 might be the combination of a synoptically induced thermal wind and a sea breeze, and the large number of similarities with sea-breeze dynamics (diurnal modulation, inertial rotation, scale of the adjustment of the order of the Rossby radius, etc.) make us confident that such a concept might be relevant.

In CN10, the jet maximum is reached at about 1800 local solar time (LST) (Figures 4 to 6 of CN10) and is most probably a combination of the ambient southwesterlies forced at synoptic scales, with V0 ∼ 5–10 m s−1, and the more local sea breeze, typically ∼ 5 m s−1. The time of the jet maximum occurs 2–5 h after the maximum land–sea temperature difference or pressure contrast (Figure 7 of CN10). In CN10's numerical investigation of one particular case, the enhanced along-coast wind basically follows the tilted isentropes similarly to Figures 3 and 4. The value of the isentrope tilt ranges between about 2 and 5 × 10−3 (Figures 15 and 16 of CN10). The vertical shear of the horizontal wind Λ ∼ 4–5×10−3 s−1 (Figures 13 and 14 of CN10), and the thermal stratification N ∼ 10−2 s−1 (tephigrams in Figures 11, 15 and 16 of CN10). For the region surrounding New York City, f ∼ 9.5 × 10−5 s−1, so the scaling variables are

  • equation image

with z0 ∼ 0.3 km, therefore equation image–0.8. In our model, the tilt of the isentropes is Λf/N2 ∼ 6.0 × 10−3 s−1, which is larger than the tilt simulated in CN10 (between 2 and 5×10−3), but still within an acceptable range considering the simplicity of our linear approach.

There is also a good qualitative comparison with the wind hodographs (Figures 6 and 12 in CN10) (Figure 5). Indeed, the wind hodographs in CN10 display a clockwise rotation in agreement with the present theory. The jet maximum is reached at ∼1800 LST, which in non-dimensional time would be equation image. In our model, equation image is in advance and π/2 (i.e. 6 h) out of phase with equation image. Conversely equation image and equation image are in phase, so that at sunrise and sunset (equation image and equation image), the breeze blows parallel to the coast with high pressure to the right. The jet maximum is equation image. If V ≫ (u,v), then the jet maximum is obtained when the synoptic wind jet blows in the same direction as the equation image-component of the breeze, i.e. from the southwest. This corresponds to high pressure over the sea at equation image, consistent with the CN10 observations. The effect of the shear, equation image, on the hodograph cannot be assessed quantitatively. Qualitatively, the wind measurements in CN10 are collected inland near the shore. The climatological hodograph (Figure 6 in CN10) corresponds to a jet maximum of about 10 m s−1, whereas the hodograph of the case-study (Figure 12 in CN10) corresponds to a jet maximum of about 15 m s−1. We cannot directly infer the value of the wind shear equation image (except for the CN10 numerical simulation where equation image–0.8), but we can postulate that the stronger the jet maximum, the stronger the shear. The climatological hodograph displays typical diurnal wind variation of about 7 m s−1 (between 5 and 12 m s−1; Figure 6 in CN10), whereas the magnitude of the variations reach 10 m s−1 for the strong jet case-study (between 2 and 12 m s−1; Figure 12 in CN10). This difference, also predicted theoretically in Figure 5(b), might be attributed to the spatial shift of the sea breeze inland and the sea-breeze intensification with shear onshore, as discussed in the previous section. Even though this interpretation must be taken with care (because of measurement uncertainties, typically of the order of 1 m s−1), the behaviour of the observed hodographs might be an indirect validation of the inland shift of the breeze in the presence of a superimposed synoptic along-shore jet.

Although the similarities between CN10 observations and our linear model reveal a broad agreement with respect to some of the basic flow features, our simple model assumptions limit a more quantitative one-to-one correspondence. Indeed, there are significant differences between CN10 observations and our model:

  • the observed/modelled evolution of the near-surface jet is tightly correlated with the evolution of a meridional pressure gradient, which cannot be included in the 2D linear model;

  • the observed jet maximum occurs about 2 to 5 h later than the pressure gradient maximum, while they are in phase in the linear model.

Regarding this last aspect, in our model, equation image and ∂ϕ/∂x are in phase, so that the pressure gradient maximum should be found when equation image is maximum (i.e. at sunset, equation image). In CN10, the jet maximum occurs about 2 to 5 h later than the pressure gradient maximum. So, the phase lag between the jet maximum and the cross-shore pressure or temperature gradient is apparently not consistent with our theory. The explanation for this difference could be twofold: (i) the absence of a cross-shore wind which can interact with the sea-breeze, and (ii) the absence of friction in our model which is known to play a major role on the phasing between the sea-breeze cell and surface heating (R83). The effect of a cross-shore background wind is geometrical, and results from the fact that the wind speed is a maximum when U0u + V0v is a maximum if U00. Taking for u and v the values given by the current theory with U0 = 0, and since they are then in quadrature, one finds a phase lag χ given by χ = tan−1U0|u|/V0|v|. Considering a 10 m s−1 offshore synoptic jet making an angle of 45° with the shore to the left, one finds a jet maximum occurring about 2 h 45 min after the maximum cross-shore pressure gradient, which is not inconsistent with CN10, considering the shape of the concave coast line where the cross-shore pressure gradient as well as the wind components are measured. Accounting for friction, by adding a linear friction term (R83; Drobinski and Dubos, 2009), i.e. (Fu,Fv,Fw,Fb) = −ν(u,v,w,b) (where ν is a friction coefficient), is equivalent to the transformation ∂/∂t [RIGHTWARDS ARROW] ℒ = ∂/∂t + ν. To investigate the effect of friction, we compute the circulation budget of the breeze flow:

  • equation image(23)

The vertical branch may be neglected in the hydrostatic approximation. The evolution of C is obtained from Eqs (5) and (23), so

  • equation image

where FV, PG are the Coriolis and pressure gradient contributions. The pressure gradient term PG is directly related to the buoyancy through the hydrostatic approximation (PG = B in R83 and Drobinski and Dubos, 2009). Following the methodology of R83, one can easily show that the circulation budget is not affected by the presence of the thermal wind-induced shear Λ, and is the same as in R83 (which is not surprising since Eq. (11) is identical to Eq. (14) of R83). We can thus show that the breeze circulation and heating display a phase lag

  • equation image

(for f2 + ν2> ω2 since f > ω) (R83). Similarly, the phase lag between buoyancy or pressure gradient and heating is χ2 = tan−1 (ω/ν) (R83). The Coriolis term FV thus displays a phase lag χ1 with respect to the pressure gradient PG, FV being a maximum after the pressure gradient maximum is reached. Figure 7 displays phase lag χ1 of the Coriolis force relative to pressure gradient as a function of the linear friction parameter ν for f = 10−4 s−1. It effectively shows that friction induces a lag between FV and PG, and thus between v and the pressure gradient ∂ϕ/∂x. Figure 7 also shows that χ1 quickly tends to a value of about π/4, i.e. 3 h, which is close to the value observed by CN10. At this stage, it is difficult to conclude which is the dominant process that can explain the phase lag between the pressure gradient maximum and the jet maximum. A full nonlinear modelling system in an idealized framework would be more suited to address this issue.

thumbnail image

Figure 7. Phase lag χ1 of the Coriolis force relative to pressure gradient as a function of the linear friction parameter ν for f = 10−4 s−1.

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5. Conclusion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The linear model
  5. 3. Solutions for prescribed heating
  6. 4. Similarities to and differences from observations
  7. 5. Conclusion
  8. Acknowledgements
  9. References

This article investigates the linear dynamics of the sea breeze in a thermal wind. It shows that the breeze cell modified by the along-shore jet has a tilt equation image. When equation image, the tilt of the breeze cell becomes equal to the slope of the background isentropes (which are in thermal wind balance with the wind shear). The critical slope of the sea-breeze tilt thus corresponds to the slope of the isentrope tilt. The present theory also predicts a spatial displacement between the heating pattern and the sea-breeze response, as the u disturbance moves onshore with increasing the thermal wind shear equation image.

The present linear theory is also used to interpret measurements made in the vicinity of New York City in the presence of a sustained synoptic southwesterly jet which occurs predominantly during the spring and summer (more than two events per month; CN10). There are very consistent results between the observations and the theory: the enhanced along-coast wind basically follows the tilted isentropes with a similar isentrope tilt, the clockwise-rotating wind hodograph showing the jet maximum peaking at 1800 LST (i.e. sunset at equation image) is also predicted by the theory. The phase lag between the jet maximum at 1800 LST and the cross-shore pressure gradient maximum about 2–5 h earlier is less straightforward, but can be attributed to the angle made by the synoptic jet with respect to the coastline and/or the presence of friction. However, there is a need to quantify the contribution of the two processes, which is outside the scope of this study and is left for future work. The inland spatial shift of the breeze might also be indirectly confirmed by the coastal inland wind observations of larger diurnal amplitude for a stronger synoptic jet (thermal wind shear).

Finally, all the comparisons that can reasonably be made between such a simple model and a very complex reality show a broad agreement of some of the basic flow features. However a deeper modelling study, where the restrictions can be relaxed one at a time, is needed to include some key aspects of the cases observed in the ‘real atmosphere’ as, for instance, an explicit expression of the meridian pressure gradient. This is left for future work.

Acknowledgements

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The linear model
  5. 3. Solutions for prescribed heating
  6. 4. Similarities to and differences from observations
  7. 5. Conclusion
  8. Acknowledgements
  9. References

We are thankful to Brian Colle for fruitful discussions and to the two anonymous referees who helped to improve the manuscript significantly.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The linear model
  5. 3. Solutions for prescribed heating
  6. 4. Similarities to and differences from observations
  7. 5. Conclusion
  8. Acknowledgements
  9. References