The Dynamics of Heat Lows Over Flat Terrain

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Q. J. R. Meteorol. Soc. 00: 1–14 (2008) Published online in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/qj.000

The dynamics of heat lows over ﬂat terrain

Thomas Spenglera∗ and Roger K. Smithb b Institute for Atmospheric and Climate Science, ETH Zurich, Zurich, Switzerland b Meteorological Institute, University of Munich, Munich, Germany

Abstract: The numerical model for a heat low developed by R´acz and Smith (1999) is extended to include a representation of radiative heating and cooling. The model is run with a higher horizontal resolution than the original version and is used to investigate further dynamical aspects of the structure and evolution of a heat low over a sub-continental or continental scale circular island surrounded by sea. Of particular interest is the diurnal and day-to-day evolution of the upper- and lower-level circulations and the degree of balance that exists in these. The heat low has a minimum surface pressure in the late afternoon or early evening following strong solar heating of the land, but the cyclonic circulation is strongest in the early morning hours following a prolonged period of low-level convergence. The low-level convergence is associated with the sea-breeze and later with the nocturnal low-level jet. The heat low is surmounted by an anticyclone, the development of which is closely tied to the outﬂow branch of the sea breeze. The anticyclone has a much smaller diurnal variation than the heat low and, unlike the heat low is largely in balance, except in the region affected by the upward-propagating gravity wave induced by the inland-penetrating sea breeze. There is a strong analogy to certain aspects of tropical cyclones, which have a warm core, a shallow unbalanced boundary layer, and which are surmounted also by an anticyclone. Copyright c 2008 Royal Meteorological Society

KEY WORDS Heat low; heat trough, thermal low, thermal trough, radiation scheme, balanced ﬂow

Received April 14, 2008; Revised ; Accepted

1 Introduction the heat low in specific regions (Leslie and Skinner 1994, Alonso et al. 1994, Portela and Castro 1996, Hoinka and Heat lows or heat troughs are prominent climatological Castro 2003). features of many arid land areas of the world during the RS99 investigated the evolution and structure of a warmer months, especially at low latitudes where insola- heat low over a square area of flat land surrounded by sea, tion is strong. These areas include northern and southwest- showing in particular the evolution of the relative vorticity ern Africa, West Pakistan and northern India, the Qinghai- and potential vorticity distributions. Their main findings Xizang plateau in China, southwestern North America, may be summarized as follows: Saudi Arabia, the Iberian Peninsula and northwestern and northeastern Australia. A list of pertinant references is • The heat low has a minimum surface pressure in given by Rácz and Smith (1999, henceforth RS99). These the late afternoon or early evening following strong systems are sometimes referred to as thermal lows or insolation of the land, while the relative vortic- thermal troughs, respectively, and they are shallow distur- ity is strongest in the early morning hours follow- bances, generally confined below 700 mb. The distinction ing a prolonged period of low-level convergence. between a heat low and a heat trough as seen in a surface Thus the heat low is not approximately in quasi- isobaric chart has no dynamical significance and we shall geostrophic balance. use the terms interchangeably. The former has at least one • The low-level convergence is associated with the closed isobar in the chart while the latter has the form of sea-breeze and later with the nocturnal low-level jet. an open wave, a distinction that may depend on the isobar • Although a cyclonic vortex, the heat low is char- spacing in use. Both types of disturbance may be thought acterized by an anticyclonic potential vorticity of as low-level cyclonic relative vorticity maxima which anomaly relative to its environment on account of are linked to horizontal gradients of diabatic heating. the greatly reduced static stability in the convec- Over the last two and a half decades there have been tively well-mixed boundary layer; however, the hor- several modelling studies of heat lows, some in idealized izontal gradient of surface potential temperature flow configurations (Leslie 1980, Fandry and Leslie 1984, over the land outwards from the centre corresponds Kepert and Smith 1992, Gaertner et al. 1993, Adams with a cyclonic potential vorticity anomaly at the 1986, 1993, RS99, Reichmann and Smith 2003, Spengler surface. • et al. 2006, Zängl and Chico 2006) and others related to As a result of the reduced static stability in the mixed layer, the horizontal components of relative vorticity and horizontal potential temperature gradi- ∗Correspondence to: Thomas Spengler, Institute for Atmospheric and Climate Science, ETH Zurich, Universitätsstr. 16, CH-8092 Zürich, ent make a non-negligible contribution to the poten- Switzerland. Email: [email protected] tial vorticity in certain flow regions.

Copyright c 2008 Royal Meteorological Society Prepared using qjrms3.cls [Version: 2007/01/05 v1.00] RS99 investigated also the effects of differing sea 2 Model description area, land area and Coriolis parameter on various aspects of the heat low, but gave little attention to the overlying The numerical model is a modified version of the hydro- anticyclone. Nevertheless, the predictions of their model static primitive-equation model described by RS99, the main changes being the implementation of a radiation have been supported by recent observations of the Saharan scheme, a relaxation scheme for the horizontal bound- heat low. They go some way also to explaining the ary conditions and ten additional upper model levels with observed diurnal cycle of the West African monsoon increased horizontal diffusion to reduce reflection of grav- circulation (Parker et al. 2005 and references) as well ity waves from the top of the domain. Other changes are as that in the heat trough over northeastern Australia those inccorporated by Spengler et al. (2005), who used (see e.g. Preissler et al. 2002). In a subsequent paper, 4 a bi-harmonic (∇ ) diffusion, and higher horizontal reso- the two authors extended their model to examine the lution (25 km instead of 100 km) to provide an improved effects of orography on heat low formation and structure representation of the sea-breeze circulation. In contrast to (Reichmann and Smith, 2003, henceforth RS03). RS99 we consider flow over a circular island instead of In the RS99 model, the cycle of diabatic heating the square one. and cooling of the atmosphere was accomplished using The model is formulated in sigma-coordinates, 1 the Mellor-Yamada 2 4 parameterization scheme for the (x, y, σ),onanorthern hemisphere f-plane centred at o boundary-layer (Mellor and Yamada 1974) in association 20 N latitude (here σ = p/ps, p is the pressure, and ps the with surface heating or cooling. After long integration surface pressure). The equations are expressed as finite- times, this scheme leads to unrealistic mixed-layer depths differences on a domain with 40 interior σ-levels. The because the only mechanism for cooling to oppose the latter are unequally spaced to provide a higher vertical net heating of the atmosphere is that which occurs in resolution in the boundary layer. The vertical grid is stag- a shallow layer adjacent to the surface. In an effort to gered with the horizontal velocity components, geopo- remove this limitation, RS03 incorporated the radiation tential, and temperature stored at σ-levels and the verti- scheme proposed by Raymond (1994, henceforth R94) in cal sigma velocity, σ˙ = Dσ/Dt, stored half way between the model. This scheme is based on a grey atmosphere σ-levels. Here D/Dt is the material derivative. A non- approximation and allows the atmosphere to cool, thereby staggered Arakawa A-Grid is used in the horizontal direc- preventing the long-term growth of the mixed layer. tions with all variables stored at each grid point. There are 199 × 199 grid points in the horizontal with a grid spacing Recently, Spengler et al. (2005) extended the RS99 of 25 km giving a total domain size 4950 km west to east model to investigate the effects of simple basic flows on and south to north. This is larger than the 4000 × 4000 km the dynamics of heat lows. During the course of that study, square domain of interest and is chosen to minimize the the first author discovered some errors in the model code influence of the boundaries on the solution. and in the radiation scheme implemented by RS03. In this paper we implement a corrected version of the 2.1 Boundary conditions radiation scheme into a higher resolution version of the RS99 model and describe some interesting new features of Following the method proposed by Davies (1983) a relax- the solutions. In particular we investigate the structure and ation scheme is applied as horizontal boundary condi- evolution of the anticyclonic circulation that forms above tion on the velocity components, temperature, and surface the heat low as well as the degree to which the flow is pressure. These fields are relaxed to the initial fields in in balance. We plan to revisit the problem investigated by a quiescent atmosphere in radiative equilibrium. At the RS03 in a subsequent paper. upper boundary (σ =0) σ˙ =0. By itself, this condition In section 2 we review briefly the model formula- would lead to the reflection of gravity waves at the top of the model domain. In order to reduce this reflection we tion, highlighting the differences from the original model. increase the horizontal diffusion in the top 10 model levels In particular, in subsection 3 we discuss the radiation by a factor of 10. The increase is linear with a log p pro- scheme proposed by R94 with our correction and exten- 10 file from 7 hPa to 0.7 hPa and was tested to sufficiently sion thereto. We begin in section 4 with a brief comparison reduce the effects of reflection. At the lower boundary of simulations with and without the radiation scheme and (σ =1), σ˙ =0 and the sea-surface temperature is held go on to describe the diurnal evolution of the heat low constant. Other conditions are specified by the boundary- and the anticyclone above it during the mature stage of layer formulation described in the appendix to RS99 and the calculation with the radiation scheme included. In this the formulation of diabatic processes considered next. section we explore also the effects of different island radii and comment on the relation of the present calculations to those of RS99. In section 5 we investigate the degree 2.2 Diabatic processes to which the flow is in gradient wind balance. Section 6 The temperature of the ground over land during many examines the spin up of the heat low and upper anticyclone diurnal cycles of heating and cooling is diagnosed from during the first six days of the calculation and section 7 a surface heat balance condition. For maximum simplic- considers briefly certain similarities of heat lows to tropi- ity, there is no moisture in the model. A heat diffusion cal cyclones. The conclusions are presented in section 8. equation is solved at three levels in the soil to compute the

Copyright c 2008 Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: 1–14 (2008) Prepared using qjrms3.cls DOI: 10.1002/qj heat ﬂux into or out of the ground. The surface parameters are detailed in Table A1 in the appendix. The version of the scheme used has been veriﬁed by Tory (1997) against data for days 33 and 34 of the Wangara boundary-layer experiment (Clarke et al., 1971).

3 Radiation scheme The radiation scheme is a modified form of the simple scheme proposed by R94 for solar and long-wave radiative transfers. In this scheme, the atmosphere is assumed to radiate as a grey body in the infra-red regime, i.e. there is no distinction in emission or absorption between different wave lengths in this regime. Absorption of solar radiation is based on a given profile adapted from Manabe and Strickler (1964), whereas the long-wave radiative transfer is calculated utilizing the Schwarzschild-Schuster approximation. The equivalent potential temperature tendency is Figure 1. Vertical plot of daily integrated temperature tendencies given by Eq. (1) in R94, i.e. −1 in K day . Labels indicate contributions from infrared (IR), solar dθe (solar), boundary layer parameterisation (BLparam), and total. = L + R, (1) dt where θe is the equivalent potential temperature, L repre- where t and s indicate the values corresponding to the sents a relaxation to a background state of the atmosphere, troposphere or stratosphere, respectively. The temperature a term omitted in the present study, and R represents the changes due to absorption of solar radiation with the the radiative heating rate given by parameters given in the table below produce heating rates θe (see Fig. 1) comparable to those of Manabe and Möller R = (Qsol + Qtherm) , (2) cpT (1961) and London (1980) for an atmosphere in radiative equilibrium. where Qsol and Qtherm are the contributions due to the absorption of solar and long-wave radiation, respectively t −5 Qmax 3.5 × 10 K/sec ≈ 3 K/day (see Eq. (8) in R94). t 3 zmax 3 × 10 m Since the present model does not include the effects t 3 zw 4 × 10 m of moisture, the equivalent potential temperature θe is s −4 Qmax 5 × 10 K/sec ≈ 43.0 K/day simply replaced by the potential temperature θ in the s 4 zmax 5 × 10 m above equations. In R94, Qsol is given by, s × 4 zw 1.8 10 m 2 z − zmax Qsol = cos αQmax exp − (3) The values for Qmax may appear rather large, but one zw must take into account the angle of the sun which enters where is a measure of the fraction of clear sky, cos(α) through the cos α term. In a grey-atmosphere model for thermal radiation is the cosine of the zenith angle of the sun and Qmax is the maximum heating rate. We use the same formula d + − here with =1because the model is dry. The units for Qtherm = − I − I , (6) −1 dz Qmax given by R94 are in [K s ], which is not in accordance with Eq. (2) where the units are [W m−2]. The where I+ and I− are the upward and downward radi- dimensionally correct form of the potential temperature ances. With the Schwarzschild-Schuster approximation tendency equation is these quantities are given by

dθ θ θ + = R = Qtherm + Qsol. (4) dI 4 + dt cpT T = ρμ(σT − I ) (7) dz In contrast to R94 we include also a solar heating and term in Qsol to provide an idealized representation of − dI 4 − radiative absorption by ozone in the stratosphere. = −ρμ(σT − I ) (8) dz − t 2 t − z zmax where μ is the effective absorptivity of the atmosphere Qsol = cos α Qmax exp t zw in the infrared, and σ is the Stefan-Boltzmann constant. Whereas R94 assumed μ to be constant, we prescribe a − s 2 s − z zmax vertical variation to account for the higher concentration + Qmax exp s (5) zw of water vapour in the lower troposphere, which is one

Copyright c 2008 Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: 1–14 (2008) Prepared using qjrms3.cls DOI: 10.1002/qj of the main absorbing agents in the long-wave band. with a radiation scheme and 6 m s−1 without one. The Speciﬁcally we take most striking feature of the azimuthal wind ﬁelds at 1700 h is the anticyclone, which has a maximum wind speed μ(z)=μ0 + μ1 exp(−z/H), (9) of over 12 m s−1 at heights between about 2 and 3 km

−4 2 −1 in the case with a radiation scheme and a maximum of where H = 7000 m. μ0 =4.0 × 10 m kg represents over 20 m s−1 at heights between about 3 and 4 km in the a background absorptivity and the second term with μ1 = −4 2 −1 case without a radiation scheme. This anticyclone feature 3.0 × 10 m kg depicts the varying contribution from was not discussed by RS99, who focussed on the low level water vapour. Substituting Eqs. (7) and (8) into Eq. (6) cyclonic flow. Interestingly, this anticyclone has a signa- yields ture throughout the depth of the troposphere, especially in the case with a radiation scheme, although it decreases in − 4 − + − − Qtherm = ρμ 2σT I I . (10) strength with height in the upper troposphere consistent with slightly cooler air overlying the heat low. In Eq. (9) in R94 the minus sign on the right-hand-side of the corresponding equations (6) and (10) is missing. This leads to an incorrect sign for the temperature tendency in 4.2 Diurnal variation Eq. (1). For example, a reduction of the upward radiance + The last two panels of Fig. 2 show the cross-sections of in a specific layer (dI /dz < 0) should cause an increase potential temperature and tangential wind speed at 0500 in temperature in the same layer (dθ/dt > 0). + h, a little before sunrise, on day 11 in the case with The lower boundary condition for I allows for a a radiation scheme. By this day the flow has reached radiative transfer from the ground into the first atmos- a mature stage with little variation from one diurnal pheric model layer. Hence an upward radiative transfer cycle to the next, although without a radiation scheme can be calculated for the lowest atmospheric layer as sug- gested by R94. the mean mixed-layer depth continues to increase (figure Since the proposed upper boundary condition, I − = not shown). Comparing these panels with the two above 0, at the top of the model domain in R94 resulted in strong shows that at 0500 h, the low-level cyclonic circulation has increased markedly with the maximum wind speed heating above about 5 km in our calculations, we replaced −1 the condition I − =0 by dθ/dt =0 at the top of the exceeding 14 m s , whereas in the case without a radiation scheme (not shown), the maximum just exceeds model domain, i.e. we assume radiative equilibrium. This −1 assumption is motivated by the high degree of radiative 18ms . In contrast the upper anticyclone exhibits balance in the mid-stratosphere. However, to apply this relatively little diurnal variation, the maximum winds condition we need to define the temperature at the top of remaining approximately the same during the 12 h to 1700 the model domain, which was chosen to be 230 K at about h. 0.7 hPa. The following discussion is based on the mature stage of the calculation (day 11) with the radiation scheme. Figure 3 shows vertical west-east cross sections of the 4 Results radial velocity component and the vertical velocity in the late afternoon (1700 h) and a little before sunrise (0500 4.1 Impact of the radiation scheme h) in this stage. The most prominent feature of the radial The impact of the new radiation scheme may be inferred velocity at 1700 h (Fig. 3a) is the shallow sea-breeze from a comparison with a calculation without a radiation inflow near the coast and a weaker offshore component scheme in which the initial temperature sounding is cho- above the sea breeze extending to about 3 km in height. sen to be the same as for all the calculations presented There is generally weak convergence over the land. The here, i.e. a temperature profile which is in radiative equi- most prominant feature of the vertical velocity field at librium over sea. The impact is clear from the first four this time is the region of strong ascent near the sea-breeze panels of Fig. 2, which show cross-sections of potential front and the weaker subsidence offshore (Fig. 3b). During temperature and tangential wind speed in the late after- the evening and early morning the sea-breeze evolves into noon (1700 h) on day 11 in calculations with and without a an even shallower nocturnal low-level jet that, by 0500 radiation scheme. The time shown is in the late afternoon h, extends over much of the land area (Fig. 3c). Unlike when the effects of insolation are near a maximum. The the sea breeze, the jet is not accompanied by significant most striking difference in the potential temperature fields vertical velocities at this time (Fig. 3d). is the depth of the mixed layer as characterized by the The instantaneous fields shown in Figs. 3 do not pro- depth of the region of uniform potential temperature at the vide a very complete view of the diurnal evolution because centre of the domain. In the calculation without a radiation the sea breeze front initiates vertically-propagating gravity scheme, this depth is around 6 km, which is unrealisti- waves. These are very evident in high-temporal-resolution cally large. With a radiation scheme, the depth is much (10 min output) animations of the radial and vertical more realistic at just over 4 km, a value that is typical velocity fields. For this reason, instantaneous fields of of inland soundings across central Australia, for exam- the vertical velocity, in particular, cannot be interpreted ple (compare Fig. 2a,c). The low-level winds are cyclonic in terms of buoyancy forcing on time scales of many and relatively light at this time, not exceeding 4 m s−1 hours. Therefore it turns out to be instructive to examine

Copyright c 2008 Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: 1–14 (2008) Prepared using qjrms3.cls DOI: 10.1002/qj (a) (b)

(e) (f)

Figure 2. Vertical cross-sections of (a) potential temperature (contour interval 3 K) and (b) the tangential wind component (contour interval 2ms−1) at 1700 h in the mature stage of the calculation (day 11) without a radiation scheme. Panels (c) to (f) show the corresponding cross-sections for the calculation with the new radiation scheme: (c) and (d) at 1700 h, and (e) and (f) at 0500 h. Only half of the domain is shown in the case of the wind ﬁelds. The thick black line along the abscissa shows the land.

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Figure 3. Vertical cross-sections of (a) radial wind speed (contour interval 1 m s−1) and (b) the vertical velocity (contour interval 2 cm s−1) at 1700 h in the mature stage of the calculation (day 11) with the new radiation scheme. Panels (c) and (d) show the corresponding cross-sections at 0500 h. The thick black line along the abscissa shows the land. time-averaged fields off all the flow components during during the afternoon, which leads to a significant weaken- selected periods of the day. Vertical cross-sections of the ing of the cyclone, and the formation of a strong low level mean fields of the tangential wind component between jet at night, which re-amplifies the cyclone. In contrast, midnight and 0600 h and between noon and 1800 h are there is little diurnal variation in the mean strengths of the shown in Fig. 4. The cross-section between midnight and upper anticyclone during these periods. 0600 h is similar to that at 0500 h (Fig. 2f) and the one Figures 5 and 6 show similar cross-sections of the between 0600 and noon (not shown) is only slightly dif- radial and vertical velocity components, respectively, ferent with the low-level cyclone being a little weaker in averaged over six hour periods commencing at midnight. magnitude than in the early morning mean. Likewise, the These cross sections are sufficiently different that we show cross-section between noon and 1800 h is similar to that at all four time periods. In the two morning periods there is 1700 h (Fig. 2d) and the one between 1800 h and midnight radial convergence in a relatively shallow layer over land (not shown) is only shightly different with the low-level with a deeper layer of outflow above (Fig. 5a, b). Above cyclone being a little stronger in magnitude than in the this is a layer of weak outflow that slopes upwards and afternoon mean. Thus, in a mean sense we can distinguish outwards with increasing radius. The vertical velocity at between distinct patterns of the tangential flow component these times (Fig. 6a, b) shows mostly upflow over the land between midnight and noon, when the low-level cyclone except for an annular region surrounding the axis. This is strongest, and between noon and midnight when it is region is relatively narrow between midnight and 0600 h much weaker. These differences reflect the strong turbu- and somewhat broader between 0600 h and noon. There is lent mixing of momentum in the mixed layer over land generally subsidence over the sea in a region that slopes

Copyright c 2008 Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: 1–14 (2008) Prepared using qjrms3.cls DOI: 10.1002/qj (a) (b)

Figure 4. Vertical cross-sections of the 6-hourly mean tangential wind component between (a) midnight and 0600 h and (b) noon and 1800 h in the mature stage of the calculation (day 11) with a radiation scheme. (contour interval 2 m s−1) The thick black line along the abscissa shows the land. downwards from mid-tropopsphere to sea-level near the the diurnal range of pmin is about 2 mb. The pressure at coast. large radius over the sea, pR, varies only slightly during The most prominent feature in the afternoon (noon the diurnal cycle. to 1800 h) is the dipole pattern of low-level inflow with outflow above it near the coast (Fig. 5c). This pattern 4.3 Larger and smaller islands marks the sea-breeze circulation (compare with Fig. 3a) It is of interest to know how the foregoing results depend and is accompanied by a strong dipole of ascent and on the size of the island. Therefore, in addition to the subsidence seen in Fig. 6c). Elsewhere there is mostly 800 km radius island considered so far, we repeated the ascent over the land and subsidence at most levels behind calculations for island radii of 400 km and 1200 km, the sea breeze front. In the evening (1800 h to midnight) the latter being of continental scale such as Australia. there is strong low-level convergence over the land and Figure 7 shows the dependence of the parameters, vmax, mostly outflow above about 2 km (Fig. 5d). The enhanced vmin, umax, umin, pmin and pR on the size of the island. inflow/outflow dipole associated with the sea breeze is The most notable differences are between the smallest still apparent, but there is a second inflow maximum at island and the others. The maximum tangential wind a smaller radius associated with the developing nocturnal during the afternoon is much weaker than in the other two low-level jet. Now there is mean ascent at most levels over and the diurnal mean pressure is higher by about 3 mb. the land, except just behind the decaying sea breeze (Fig. Nevertheless, the diurnal pressure variation is similar in 6d). all three cases. Figure 7 summarizes the diurnal variation of the Figure 7d compares the diurnal pressure variation for strength of the heat low, as determined by the maximum the three island radii with the mean observed variation at tangential wind speed (vmax) and the minimum surface five surface stations between about 21oS and 24oS over pressure (pmin), and of strength of the upper-level anticy- central Australia obtained during a three week period in clone, as determined by the maximum anticyclonic wind September/October 1991 during the Central Australian speed (vmin) (refer to the curves labelled 8). It shows Fronts Experiment (Smith et al. 1995). It is interesting also the variation in the strength of the meridional circula- that the pressure variation in the calculation accounts for tion, as measured by the strength of the maximum inflow at least half of the observed variation and the maximum (umin) and outflow (umax), and the surface pressure at occurs at about the same time or one to two hours later, the domain boundary, pR. While the minimum surface depending on the size of the island. On the other hand it pressure occurs at 2000 h, the maximum tangential wind accounts for significantly less than half of the observed speed is attained between 0500 h and 0600 h and mini- minimum and the minimum is about four hours later than mum at 1600 h. Note that the diurnal range in vmax has that observed. That is not to suggest that the agreement an amplitude of 12 m s−1. In contrast the strength of the with observations is poor as the observed tidal variation upper anticyclone shows no diurnal oscillation, although is associated also with a gravity wave induced by the it increases slightly by 1.5 m s−1 during day 11, showing radiative heating of the upper atmosphere during the day that the mature state on this day is still slowly evolving. and cooling during the night, a heating and cooling pattern The maximum value of pmin occurs just before noon, and that circles the globe once a day (see e,g. Chapman and

Copyright c 2008 Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: 1–14 (2008) Prepared using qjrms3.cls DOI: 10.1002/qj (a) (b)

Figure 5. Vertical cross-sections of the 6-hourly mean radial wind component between (a) midnight and 0600 h, (b) 0600 h and noon, (c) noon and 1800 h, and (d) 1800 h and midnight in the mature stage of the calculation (day 11) with a radiation scheme. (contour interval 1 ms−1) The thick black line along the abscissa shows the land.

Lindzen, 1970). In our model, the heating and cooling 4.4 Comparison with RS99 pattern does not vary with longitude. A quantitative comparison of the present calculations In the calculation for an island radius of 1200 km, with those of RS99 is not warranted because here the the diurnal mean pressure is only slightly lower than that environmental sounding is determined by the radiation for an island radius of 800 km. The anticyclone is about scheme and is therefore different from that used by RS99. 20% weaker in the case of the smaller island, but is similar Furthermore, RS99 used a square island instead of the in strength in the other two cases. The most apparent circular one used here. Because of the higher horizontal difference in the vertical cross sections (not shown) is resolution and more selective horizontal diffusion in the that the sea breeze reaches the centre of the domain in present model, the sea breeze circulation is much more the case of the smallest island, but not in the other two. pronounced, otherwise the basic ﬂow behaviour is similar In the larger island cases, the low-level tangential wind in both models. The main calculation presented by RS99 component has two maxima in the early morning, one had a land area of 600 × 600 km, comparable with the size associated with the decaying sea breeze and the other of the Iberian Peninsula, providing the opportunity for the associated with the nocturnal low-level jet over the island, sea breezes to reach the domain centre by evening. This which develops independently where there is a horizontal happens here only in the calculation with a land radius pressure gradient (Fig. 8b). As the island size decreases, of 400 km. The RS99 calculation showed also an upper the separation between these two maxima decreases and level anticyclone, but its strength is somewhat weaker than for the smallest island there is only one maximum (Fig. in the main calculation analyzed here, consistent with the 8a). island size inﬂuence noted above (see Fig. 7a).

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Figure 6. Vertical cross-sections of the 6-hourly mean vertical velocity component between (a) midnight and 0600 h, (b) 0600 h and noon, (c) noon and 1800 h, and (d) 1800 h and midnight in the mature stage of the calculation (day 11) with a radiation scheme. (contour interval 2cms−1) The thick black line along the abscissa shows the land.

5 Balanced diagnostics cross-sections of this quantity are shown in Fig. 9.At 1700 h, the main regions of imbalance are associated with The relatively small diurnal variation in the upper-level the sea-breeze flow and the low-level inflow over land. anticyclone evident in panels (d) and (f) of Fig. 2, in Fig. Other small regions where the radial pressure gradient 4 and in Fig. 7a suggest that this feature may be close force deviates from its balanced value by more than 2 −2 to gradient wind balance. One consequence of balance ms are associated with gravity waves as indicated by would mean that the slow evolution of the vortex could be animations of the fields at 10 min. intervals. At 0500 h, described using a Sawyer-Eliassen-type equation with the the main regions of imbalance are associated with the low- heating-cooling distribution determined by the radiative level jet and with a vertically-propagating gravity wave heating/cooling (see e.g. Montgomery et al. (2006) for located above the jet. Animations of vertical cross sections the cace of a tropical cyclone vortex). It would seem of various quantities plotted at 10 min intervals suggests pertinent, therefore, to examine the balanced aspects of that the lack of balance associated with the sea breeze the flow in more detail. This problem is simplified here is an important feature of the spin-up of the heat low as because the model is hydrostatic, ensuring that the flow discussed in the next section. evolves in hydrostatic balance. Thus any imbalance must reside in the radial momentum equation. It is of interest, therefore, to calculate the net radial force, i.e. the part of the radial pressure gradient force that is not balanced by the sum of the centrifugal and Coriolis forces. Vertical

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Figure 7. Time series of the magnitude of the maximum (solid curves) and minimum (dashed curves) (a) tangential velocity, vmax and vmin (m s−1), (b) radial velocity, umax and umin (m s−1), and (c) minimum pressure, pmin (mb) in the mature stage of the calculation. The designation 4, 8, and 12 refers to island radii of 400 km, 800 km and 1200 km, respectively. The solid curves in panel (c) are the surface pressure at large radius. (d) Diurnal variation of pmin in the calculations for different island radii compared with the observed mean variation (solid curve) at stations near 20oS over central Australia in September/October.

(a) (b)

Figure 8. Vertical cross-sections of the tangential wind component (contour interval 2 m s−1) at 0400 h in the mature stage of the calculation (day 11) for (a) the small island, and (b) the large island. The thick black line along the abscissa shows the land.

6 Spin up apart during the first three days of the calculation and in Fig. 11 we show the day-to-day variation of the quanti- The relative steadiness of the upper anticyclone and its ties shown in Fig. 7. The left panels of Fig. 10 show the deep vertical extent raises interesting questions concern- pattern of inflow and outflow at 1800 h each day. The ing its evolution. To illustrate the spin-up of the anticy- maximum inflow increases slightly and whereas the max- clone we show in Fig. 10 vertical cross sections of radial imum outflow changes little during the three days, the and tangential wind components at selected times 24 h vertical extent increases significantly. We argue that this

Copyright c 2008 Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: 1–14 (2008) Prepared using qjrms3.cls DOI: 10.1002/qj (a) (b)

Figure 9. Vertical cross-sections of unbalanced pressure gradient force per unit mass (contour interval 5 × 10−5) ms−2) up to a height of 10 km at (a) 1700 h and (b), 0500 h. The thick black line along the abscissa shows the land. is the reason why the anticyclone becomes progressively diurnal variation of turbulent mixing over land compared deeper as seen in the right panels of Fig. 10 at midnight with over the sea. on each day. It is clear from these panels that the upper anticyclone is associated with the outflow branch of the 8 Conclusions sea breeze and that its vertical extent increases as the depth of the outflow region above the sea breeze increases. We have presented an improved numerical model for Animations of the fields show this association more dra- a heat low including a radiation scheme. The radiation matically than the sequence of snapshots in Fig. 10. Fig- scheme is a corrected and extended version of that pro- ure 11 shows that the maximum inflow increases slightly posed by Raymond. Its implementation leads to a more from day-to-day, a feature that is consistent with a gradual realistic depth of the daytime mixed layer. We used the growth in the maximum temperature of the mixed layer. new model to investigate aspects of the dynamics of heat This increase is accompanied by a gradual increase in the lows not previously touched upon including: maximim tangential wind speed and in the strength of the • the development of an upper-level anticyclone, and upper anticyclone, and a gradual decrease in the minimum • the degree to which the cyclonic and anticyclonic pressure. On the other hand, the mean diurnal pressure at circulations are in gradient wind balance. large radius shows little change. The maximum outflow, as opposed to its depth, does not change appreciably during We showed that the upper level anticyclone extends the first six days. through much of the troposphere, but has its maximum strength in the lower troposphere, just offshore. It exhibits relatively little diurnal variation when the heat low reaches 7 Similarities with tropical cyclones its mature stage, much less so than the low level cyclone. Heat lows fall in the same class of warm-cored vortices The anticyclone develops steadily over a period of a few as tropical cyclones with the cyclonic circulation being days and is associated with the return (offshore) branch of the sea-breeze circulation and its upward extension result- strongest at low levels and decaying with height, evening from outflow from the heat low. An important aspect tually becoming anticyclonic where the flow is radially of the evolution is the vertically-propagating gravity wave outwards. There are similarities also in the role of sur- initiated by the inland penetrating sea-breeze front during face friction in the two types of system, each having strong the afternoon and evening. Unlike the heat low, the upper inflow in a shallow boundary layer. In the heat low case, anticyclone it is largely in gradient wind balance, except in the inflow is associated with the nocturnal low-level jet. the neighbourhood of this gravity wave. The gravity wave The essential difference between the two systems is the has a significant effect on the vertical motion field at any way in which the warm core is maintained: in the heat time and time averaging is required to isolate the more low it is primarily a result of diabatic heating in a deep, steady components of the vertical motion. but dry convective boundary layer whereas in the tropical In a mean sense we can distinguish between distinct cyclone, it is a result of moist deep convection in conjunc- patterns of the tangential flow component tion with enhanced surface moisture fluxes. One essential difference is that the frictional boundary layer in the heat • between midnight and noon, when the low-level low has a large diurnal component as a result of the strong cyclone is strongest, and

Copyright c 2008 Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: 1–14 (2008) Prepared using qjrms3.cls DOI: 10.1002/qj (a) (b)

(e) (f)

Figure 10. Vertical cross-sections of the radial wind component (left panels) and tangential wind component (right panels) (contour interval 2ms−1) at selected times (a) 18 h, (b) 24 h, (c) 42 h, (d) 48 h, (e) 66 h, (f) 72 h during the ﬁrst three days of the calculation with a radiation scheme. Only half of the domain is shown in the case of the wind ﬁelds. The thick black line along the abscissa shows the land.

Copyright c 2008 Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: 1–14 (2008) Prepared using qjrms3.cls DOI: 10.1002/qj (a) (b)

Figure 11. Time series of (a) vmax, vmin (m s−1), (b) umax, umin (m s−1), (c) pmin, pR (mb), (d) θ(r =0,z =1km) during the ﬁrst six days of the calculation with the new radiation scheme.

• between noon and midnight when it is much weaker. Hoinka KP, Castro M. 2003 The Iberian peninsula thermal low. Quart. J. Roy. Meteor. Soc., 129, 14911511 These differences reflect the strong turbulent mixing of Kepert JD, Smith RK. 1982 A simple model of the Australian west coast momentum in the mixed layer over land during the after- trough, Mon. Wea. Rev., 120, 2042-2055 Leslie LM, Skinner TC. 1994 Real-time forecasting of the western noon, which leads to a significant weakening of the Australian summertime trough: Evaluation of a new regional model. cyclone, and the formation of a strong low level jet at Weather and Forecasting—, 9, 371-383 night, which re-amplifies the cyclone circulation. London J. 1980 Radiative energy sources and sinks in the stratosphere and mesosphere. Proceedings of the NATO Advanced Study Institute on Atmospheric Ozone, A. C. Aiken, Ed., Rep. No. FAA-EE-80-20, U.S. Dept. of Transportation. 703-721 Acknowledgements Manabe S, Strickler RF. 1964 Thermal equilibrium of the atmosphere with a convective adjustment. J. Atmos. Sci., 21, 361-385 We thank Günther Zängl and Jan Schween for stimulating Manabe S, Möller F. 1961 On the radiative equilibrium and heat balance of the atmosphere. Mon. Wea. Rev., 89, 503-532 discussions about some numerical and radiation issues, Mellor G, Yamada T. 1974 Hierarchy of turbulent closure models for which helped to improve the paper. planetary boundary layers. J. Atmos. Sci., 31, 1791-1806 Montgomery MT, Nicholls ME, Cram TA, Saunders AB. 2006 A vortical hot tower route to tropical cyclogenesis. J. Atmos. Sci., 63, References 355-386 Parker DJ, Burton RR, Diongue-Niang A, Ellis RJ, Felton M, Taylor CM, Thorncroft CD, Bessemoulin P, Tomkins AM. 2005 The diurnal Adams M. 1986 A theoretical study of the inland trough of northeastern cycle of the West African Monsoon circulation. Quart. J. Roy. Met. Australia. Aust. Met. Mag., 34, 85-92 Soc., 131, 2839-2860 Adams M. 1993 A linear study of the effects of heating and orography Portela A, Castro M. 1996 Summer thermal lows in the Iberian penin- on easterly airstreams with particular reference to northern Australia, sula: A three-dimensional simulation. Quart. J. Roy. Meteor. Soc., Aust. Met. Mag., 42, 69-80 122, 1-22 Alonso S, Portela A, Ramis C. 1994 First considerations on the structure Preissler M, Reeder MJ, Smith RK. 2002 A case study of a heat low and development of the Iberian thermal low-pressure system. Ann. over central Australia. Aust. Met. Mag., 51, 155-163 Geophys., 12, 457-468 Rácz Z, Smith RK. 1999 The dynamics of heat lows. Quart. J. Roy. Chapman S, Lindzen RS. 1970 Atmospheric tides, D. Reidel, Dodrecht, Meteor. Soc., 125, 225-252 Holland 200pp Raymond DJ. 1994 Convective processes and tropical atmospheric Clarke RH, Dyer AJ, Brook RR, Reid DG, Troup AJ. 1971 The Wangara circulations. Quart. J. Roy. Meteor. Soc., 120, 1431-1455 experiment: boundary layer data. Tech. Paper 19. Div. Meteor. Phys., CSIRO, Australia, 316 pp Reichmann Z, Smith RK. 2003 Terrain influences on the dynamics of heat lows. Quart. J. Roy. Meteor. Soc., 129, 1779-1793 Davies HC. 1983 Limitations of some common lateral boundary schemes in regional NWP models. Mon. Wea. Rev., 111, 2013-2020 Smith RK, Reeder MJ, Tapper NJ, Christie DR. 1995 Central Australian cold fronts. Mon. Wea. Rev., 123, 19-38 Fandry CB, Leslie LM. 1984 A two-layer quasi-geostrophic model of summer trough formation in the Australian subtropical easterlies. J. Smith RK, Reeder MJ, May P, Richter H. 2006 Low-level convergence Atmos. Sci., 41, 807-181 lines over northeastern Australia. II. Southerly disturbances. Mon. Wea. Rev., 134, 3109-3124 Gaertner MA, Fernndez C, Castro M. 1993 A two-dimensional simulation of the Iberian summer thermal low. Mon. Wea. Rev., 121, Spengler T, Reeder MJ, Smith RK. 2005 The dynamics of heat lows in 2740-2756 simple background flows. Quart. J. Roy. Meteor. Soc., 131, 3147-3165

Copyright c 2008 Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: 1–14 (2008) Prepared using qjrms3.cls DOI: 10.1002/qj Appendix: Model parameters

Table I. Model parameters.

Parameter Symbol Value Surface albedo A 0.3. 2 −2 −1 Specific heat of dry air at constant pressure cp 1004.67 m s K Thickness of the first soil layer D1 7.2 cm Thickness of the second soil layer D2 43.2 cm Thickness of the third soil layer D3 43.2 cm Coriolis parameter f 4.988 × 10−5 s−1 −2 Solar constant F0 1365 W m Gravitational constant g 9.806 m von Kármán’s constant k 0.4 13 4 −1 Diffusion parameter K0 5 × 10 m s Reference pressure p0 1000 mb Specific gas constant for dry air R 287.04 m2 s−1 K−1 o Sea surface temperature Ts 24 C o Deep-soil temperature Tcl 25 C Surface roughness length for momentum over land zol 10 cm Surface roughness length for momentum over sea zos 1cm Surface roughness length for heat over land zhl, zol 10 Surface roughness length for heat over sea zhs, zos 4.5 Thermal emissivity ε 0.98 Thermal conductivity κ 7.5 × 10−7 m2 s−1 −8 −2 −4 Stefan-Boltzman constant σb 5.67 × 10 Wm K

Tory K. 1997 The effect of the continental planetary boundary layer on the evolution of fronts. Ph. D. dissertation, Monash University, 292pp Z¨angl G, Chico SG. 2006 The thermal circulation of a grand plateau: Sensitivity to the height, width, and shape of the plateau. Mon. Wea. Rev., 134, 2581-2600

Copyright c 2008 Royal Meteorological Society Q. J. R. Meteorol. Soc. 00: 1–14 (2008) Prepared using qjrms3.cls DOI: 10.1002/qj