Examensarbete vid Institutionen för geovetenskaper
Degree Project at the Department of Earth Sciences
ISSN 1650-6553 Nr 504
Cloud Streets. A Study of the
Instability Mechanisms Giving Rise
to Boundary Layer Rolls
Molngator - En studie över hur
molnrullar uppkommer i gränsskiktet
Josefine Bergstedt
INSTITUTIONEN FÖR
Examensarbete vid Institutionen för geovetenskaper
Degree Project at the Department of Earth Sciences
ISSN 1650-6553 Nr 504
Cloud Streets. A Study of the
Instability Mechanisms Giving Rise
to Boundary Layer Rolls
Molngator - En studie över hur
molnrullar uppkommer i gränsskiktet
Josefine Bergstedt
ISSN 1650-6553
Copyright © Josefine Bergstedt
Published at Department of Earth Sciences, Uppsala University (www.geo.uu.se), Uppsala, 2020
Abstract
Cloud streets. A study of the instability mechanisms giving rise to boundary layer rolls Josefine Bergstedt
Boundary layer rolls are a rather frequent phenomena, where regions of alternating up- and downdraft motion causes clouds to form in elongated, parallel rows oriented with the mean wind direction. The clouds can be seen during certain atmospheric conditions and are often called ”cloud streets” because of their characteristic appearance. By performing a linear instability analysis, the underlying mechanisms causing the onset of boundary layer rolls has been analysed in this study. There are two governing mech- anisms that cause the boundary layer rolls to form, the thermal instability and the dynamic instability.
The thermal instability is caused by convection in an unstable airmass, while the dynamic instability usually is associated with neutral or stable conditions. The dynamic instability arise due to an inflection point in the wind profile, around which eddies develop.
In a previous study by Svensson et al. (2017), rolls were observed over the Swedish east-coast, stretching out over sea during four days; 2 of May 1997, 3 of May 1997, 17 of May 2011 and 25 of May 2011.
The aim of this study is to simulate the rolls on these four dates, analyse the underlying mechanisms and establish what type of instability that primarily causes the rolls to form. The linear stability analysis performed in this study indicate that the dynamic instability is the main mechanism giving rise to the rolls on all four studied dates. The rolls are found to arise over the Swedish mainland and are advected out over the sea. Both the orientation of the rolls and the modeled wind direction are in accordance with the observations. A qualitative agreement is found for the wavelength, the amplitude and the altitude of the rolls, when comparing the results of this study with the observations.
Keywords: Cloud streets, boundary layer rolls, dynamic instability, thermal instability, horizontal con- vective rolls, inflection point
Degree Project E in meteorology, 1ME422, 30 credits Supervisor: Johan Arnqvist
Department of Earth Sciences, Uppsala University, Villavägen 16, SE-752 36 Uppsala (www.geo.uu.se) ISSN 1650-6553, Examensarbete vid Institutionen för geovetenskaper, No. 504, 2020
The whole document is available at www.diva-portal.org
Populärvetenskaplig sammanfattning
Molngator - En studie över hur molnrullar uppkommer i gränsskiktet Josefine Bergstedt
Molngator är ett fenomen som uppstår vid speciella förhållanden i atmosfären, vilket gör att molnen arrangeras i långa, parallella rader. Denna typ av moln uppkommer när luften rör sig omväxlande up- påt och nedåt. När luften rör sig uppåt kondenseras vattenångan som finns i luften, vilket bildar moln, medan det inte bildas några moln i områden där luften rör sig nedåt. Det möjligt att analysera hur mol- ngator uppkommer genom att undersöka mekanismerna som gör att luften ömsom strömmar uppåt och ömsom nedåt på detta sätt i atmosfären. Till grund för detta arbete ligger en studie utförd av Svensson et al. (2017), där molngator observerades över den svenska östkusten vid följande fyra tillfällen: 2 Maj 1997, 3 Maj 1997, 17 Maj 2011 and 25 Maj 2011. Syftet med det här arbetet är studera orsaken till att molngator uppstod dessa dagar. Med hjälp av datamodeller simuleras och återskapas molnen för dessa fyra dagar. Resultaten tyder på att molnen bildades över fastlandet och förs sedan ut till havs med vin- dens hjälp. Resultaten från den här studien stämmer väl överens med de mätningar, observationer och slutsatser som presenteras i Svensson et al. (2017).
Nyckelord: Molngator, molnrullar i gränsskiktet, dynamisk instabilitet, termisk instabilitet, horison- tella konvetiva rullar, inflektionspunkt
Examensarbete E i meteorologi, 1ME422, 30 hp Handledare: Johan Arnqvist
Institutionen för geovetenskaper, Uppsala universitet, Villavägen 16, 752 36 Uppsala (www.geo.uu.se) ISSN 1650-6553, Examensarbete vid Institutionen för geovetenskaper, Nr 504, 2020
Hela publikationen finns tillgänglig på www.diva-portal.org
Table of Contents
1 Introduction ... 1
1.1 Appearance and occurrence of Boundary Layer Rolls... 1
1.2 Characteristics of Boundary Layer Rolls ... 3
1.3 Formation and origin of Boundary Layer Rolls ... 3
1.3.1Thermal instability 1.3.2Dynamic instability 2 Method and model setup ... 7
2.1 Method ... 7
2.2 Simulation setup ... 8
2.3 Analysis approach ... 9
3 Weather conditions and data... 10
4 Results ... 12
4.1 May 2, 1997 ... 12
4.1.1Land 4.1.2Sea 4.2 May 3, 1997 ... 15
4.2.1Land 4.2.2Sea 4.3 May 17, 2011 ... 18
4.3.1Land 4.3.2Sea 4.4 May 25, 2011 ... 20
4.4.1Land 4.4.2Sea 5 Discussion ... 24
6 Conclusions ... 25
6.1 Definition of parameters ... 29
6.2 Derivation of the equations... 29
1. Introduction
1.1 Appearance and occurrence of Boundary Layer Rolls
Boundary layer rolls (BLR) are an atmospheric phenomena, where alternating up- and down- ward vertical winds generate counter-rotating horizontal vortices (Weckwerth et al., 1996; Müller et al., 2013). These vertical motions arise due to perturbations in the temperature-, wind- and humidity fields combined (Sandeepan et al., 2013). As the air rises, the water condense and clouds form in the upward regions, while the downward regions remain cloud-free (Etling et al., 1992). A schematic drawing of the vertical motions in BLR are shown in figure 11. The vertical wind pattern is also influencing the horizontal wind field, in such a way that higher horizontal wind speed is found where the vertical motion is directed downward and lower hori- zontal wind speed in the up-draught regions (Svensson et al., 2017). Observational studies have proven that the up- and downward vertical motions play an important role in entrainment pro- cesses, mixing and redistribution of heat and moisture in the planetary boundary layer (PBL) (Weckwerth et al., 1996).
Figure 1. Schematic figure of counter-rotating boundary layer rolls, showing the up- and downward movement of the air. Longitudinal cloud bands develop approximately aligned with the mean flow. Where is the angle between the roll axis and the mean flow. λ is the wavelength. Source: (Brown, 1980).
1With rights to republish figures according to AGU policy.
BLR are fairly common in the atmosphere, occurring on a mesoscale and ranging from the ground all the way throughout the PBL (Etling et al., 1992; Kuettner, 1971; Svensson, 2018).
The frequencies of the waves range from 15 min up to 2 hours (Brown, 1972). This has been predicted by both theory and numerical simulations and has been confirmed observationally by satellite images, where it can be seen as fluctuating wind speed in the surface horizontal wind profile due to the alternating up- and down-draughts (Weckwerth et al., 1996; Svensson, 2018).
During certain conditions, the rolls may be visualised as extended, coherent cylindrical rows of clouds (Etling et al., 1992), as seen in figure 21. Due to their appearance, BLR are often called
”cloud streets”, but many other terms for the phenomena are used, e.g horizontal roll vortices, horizontal convective rolls and longitudinal rolls (Svensson, 2018).
Figure 2. Cloud streets over the coast of Georgia US. Source: Brown (1980).
The rolls may cover great areas horizontally, with lengths varying from 10-500 km (LeMone, 1973). In the vertical, the rolls are less extended, an average depth of the rolls is 0.5-2 km (Svensson et al., 2017). BLR have been observed both over land and sea, mostly during stable atmospheric conditions (Svensson et al., 2017). The most pronounced BLR take place over sea during so called ”cold air outbreak”, when cold air originating over land, is advected out over
the warmer sea (Etling et al., 1992). This gives rise to a large temperature gradient between sea and land, leading to unstable conditions and enhanced convection over the sea (Svensson, 2018).
1.2 Characteristics of Boundary Layer Rolls
During the occurrence of BLR, several parameters adopt specific values that are characteristic for BLR events, the wavelength being one of the most important parameters. The wavelength is defined as the distance between the minima or maxima of the wind speed, hence include one pair of rolls (Svensson, 2018), as seen in figure 1. The wavelength varies greatly, from 1 km as reported by LeMone (1973) up 20 km, according to Etling et al. (1992) & Müller et al.
(2013). On average, the roll wavelength lies within the range of 2-8 km (Kuettner, 1971). The wavelength is often expressed in terms of the height of the PBL instead of kilometes, the so called ”aspect ratio” (Weckwerth et al., 1997). Expressed in these terms the wavelength of the rolls is roughly 2-4 times the PBL height (Brown, 1980), but may have a wider range of 2-15 times the PBL height (Etling et al., 1992).
Another significant feature of BLR, is that the maximum vertical- and horizontal roll velocity occur at certain heights. In the vertical, the maxima is reached in the middle of the boundary layer, approximately at 0.33 times the inversion height. The horizontal maximum is found closer to the ground, at ∼0.07 times the inversion height (LeMone, 1973). Cloud streets are usually oriented with the axes aligned with the direction of the mean boundary layer wind (Brown, 1972; LeMone, 1973). While, at the inversion base, the rolls usually make an angle of -5◦ to 20◦ to the left of the geostrophic wind (LeMone, 1973).
1.3 Formation and origin of Boundary Layer Rolls
BLR develop when there is an instability in the airflow. The primary mechanisms causing BLR are either dynamic or thermal instability or more often, a combination of the two (Svensson et al., 2017) and references therein. The thermal instability arises in the convective boundary layer when an air parcel is warmer than the surrounding air and becomes positively buoyant. Hence, thermal instability is associated with unstable stratification (Svensson, 2018). The dynamic instability usually form in stable or neutral flow and is caused by an inflection point in the wind profile, around which vorticity emerge (Svensson, 2018).
Previous studies of BLR events states that there are a few criteria that must be met in order for BLR to develop. The vertical wind shear gradient is an important parameter if the rolls are caused by dynamic instability. According to Weckwerth et al. (1997) the wind shear highly impact the development of BLR. Kuettner (1971) states that the wind shear should be approx- imately 10−7-10−6 cm−1s−1, which indicates a strong shear curvature. Computer simulations show that an increase in wind shear favours three dimensional convective cells to develop into two dimensional rolls (Weckwerth et al., 1997). BLR are also associated with relatively high
wind speeds. Svensson et al. (2017) reports that the wind speed must reach a certain threshold value of at least 5.5 m/s in order for BLR to form. Weckwerth et al. (1998) argue that neither the wind speed nor the wind shear affect the onset of rolls, but acknowledge the fact that both the wind speed and wind shear influence the evolution of rolls. A a decrease in wind speed will cause the rolls to contract and have a smaller extension and evolve into open cells, (Weckw- erth et al., 1998). However, if the wind speed becomes too high, the rolls will dissipate into unorganised convection (Weckwerth et al., 1998). Weckwerth et al. (1998) argues that the most suitable parameter for determining roll formation is the buoyancy flux. In their study they find that rolls arise independently of the wind speed and the wind shear, when the average buoyancy flux exceed 0.05 K ms−1(∼ 50 W m−2).
1.3.1 Thermal instability
The stability of the atmosphere is essential in determining the onset of convection. Thermal instability arises when the air is heated from below, due to e.g surface heating, or ”cold air out- break”, when cold air is advected over warmer sea surface. The air becomes buoyant, unstable and begins to rise. Within a normal boundary layer with basic flow, the wind velocity profile is curved, i.e logarithmic, and the vorticity gradient is negative with height in the atmosphere.
This means that the vorticity is higher in a layer closer to the ground than in layers higher up.
As the warmer air rises, the vorticity of the air parcel is conserved, resulting in the rising air having a higher vorticity than the surroundings and the vorticity field is disturbed (Kuettner, 1971). Due to this disturbance, the parcels to the left of the vortex will be replaced by air from lower levels, with a higher vorticity and air parcels to the right of the vortex are replaced by sinking air from above, having lower vorticity. This will result in a total downward motion of the air, bringing the rising air back to its original position. Hence, the buoyancy forces and the vorticity forces are in equilibrium and the atmosphere is stable. Figure 32schematically show these motions in a stable atmosphere. However, there are some situations when the buoyancy force is dominant over the restoring vorticity forces and the convection becomes unstable. For this to happen, the convection has to be symmetric and the circulation perpendicular to the air- flow. This would, for example, occur if all the air parcels that are aligned parallel with the mean flow start to rise simultaneously (Kuettner, 1971, 1965). This would cause the helical motion in longitudinal rolls (Kuettner, 1971). Parallel to the mean flow there is no vorticity gradient, and hence convection parallel to the mean flow could also become unstable (Kuettner, 1971).
Based on these findings, Kuettner (1965) concluded that thermal instability in combination with vorticity forces, arising from a curved wind profile causing the onset of BLR in a moving air mass.
2With rights to republish figures according to AGU policy.
Figure 3. Schematic figure of the rising and sinking motions of the air under stable atmospheric conditions. The wind speed profile is shown to the left, as is the density profile and the local vorticity. The right panel show the conservation of vorticity as air rises or sinks and the net downward motions caused by the restoring vorticity force.
Source: Kuettner (1971)
Another parameter commonly used to determine the stability of the atmosphere is the Rayleigh number. This is a dimensionless parameter that describes the onset of free convection occur- rence. The Rayleigh number is defined as eq. (1). Where ν-kinematic viscosity, κ-thermal diffusivity, β- coefficient of thermal expansion, ∆T-temperature difference, H-vertical displace- ment (Markowski & Richardson, 2010).
Ra = g∆T βH3
νκ (1)
Molecular diffusion and thermal conductivity of the air counteract the convective motions, and acts to stabilise the air. Hence, a low Rayleigh number indicate laminar flow The flow is tur- bulent when the Rayleigh number adopt a high value and so convection is triggered only when the Rayleigh number exceed a critical value, Racrit (Markowski & Richardson, 2010). For a non-flowing air mass, experiencing no vertical wind shear, convection starts at Racrit ∼ 120 (Markowski & Richardson, 2010). The threshold value, Racrit, adopts a higher value in the presence of wind shear, often in the order of several magnitudes (Kuettner, 1971). The larger the temperature difference, the higher the Rayleigh number, an effect that is counteracted by the fact that viscosity and diffusivity tend to decrease the Rayleigh number (Markowski &
Richardson, 2010). It has been shown that as the Rayleigh number surpasses the threshold value, 2-dimensional rolls form, as this is the first stable solution of convection (Weckwerth et al., 1998). For other types of organised convection to arise, e.g cellular or three-dimensional convection, the Rayleigh number must reach a higher value (Kuettner, 1971). Therefore, BLR are the first mode of structured convection (Weckwerth et al., 1998).
The stability of the convective boundary layer (CBL) may also be expressed as the ratio between the inversion height (zi) and the surface layer parameter, the Obukhov length, defined as in eq. (2) (Weckwerth et al., 1997). In eq. (2), θv-virtual potential temperature, u0w0 and v0w0 are the mean kinematic momentum fluxes, g-gravitational acceleration, k- von Kármán constant, w0θ0-kinematic buoyancy flux, s indicate that the value is taken near the surface.
L = − ¯θv(u0w0s2+ v0w0s2)(3/4)
kgw0θ0vs (2)
Turbulence in the boundary layer may arise from either dynamical instability, caused by the vertical wind shear, or buoyancy from thermal instability. The Obukhov length involves both of these instabilities and basically describes the depth of the turbulent layer when the dynam- ical instability dominates over the thermal instability (Sandeepan et al., 2013; Markowski &
Richardson, 2010) In the literature, a plethora of restrictions for the Obukhov value during BLR is presented, but rolls seems to be most favourable during slightly unstable conditions , -zi/L <
5 Etling et al. (1992). LeMone (1973) observes rolls when 3 ≤ -zi/L≤ 10 and values as high as -zi/L ∼ 272 have been reported by Christian & Wakimoto (1989). As the instability increases in the CBL, the rolls will become less extensive. For values of -zi/L> 25 the rolls start to dissipate and evolve into less organised three dimensional convection (Etling et al., 1992; Weckwerth et al., 1998; Svensson et al., 2017). Between 25<-zi/L < 5 there will exist some sort of mixed conditions including both rolls and organised convective cells (Etling et al., 1992). Although, Weckwerth et al. (1998) also finds a value of -zi/L < 25 during their observed roll events, they claim that this feature alone is not enough to determine the onset of BLR.
1.3.2 Dynamic instability
The dynamic instability can be categorised into parallel and inflection point instability, where the last one is considered the primary cause of eddies in the atmosphere (Svensson et al., 2017).
The criterion for the inflection point instability was stated by Rayleigh in 1916 (Brown, 1980).
By studying the inviscid stability equation, Rayleigh concluded that an inflection point instabil- ity arises when two adjacent air masses move at different velocity relative to each other (Brown, 1980). This phenomena is occurring quite frequently in the atmosphere, and may appear for dif- ferent reasons, e.g when two air masses of different origin meet or when the air gets stratified due to surface heating (Brown, 1980). In fact, the flow of an air mass is altered by several pa- rameters, any distinct change in either temperature, density or humidity would affect the wind speed, wind direction or both (Brown, 1980). When two air masses move relative to each other they experience horizontal shear due to the differences in wind velocity. A shear instability will arise at the boundary between the two layers and the flow will start to change with height (Brown, 1980; Markowski & Richardson, 2010). As the flow pattern is altered, local maxima in horizontal wind shear or vertical vorticity arise in the cross-roll wind speed profile and an inflec-
tion point will form at these extremes (Svensson et al., 2017; Markowski & Richardson, 2010).
In conjunction with these extremes, energy is transformed from the mean flow into vorticity and waves start to develop at the inflection point (Markowski & Richardson, 2010).
Stratification strongly influences the behaviour of the inflection point instability. The in- stability growth rate depends on the local Richardson number at the inflection point height.
Most favourable conditions for dynamically caused BLR are neutral conditions (Svensson et al., 2017). Stable stratification has a dampening effect on the instability and for a critical value of Ri>0.25, the instability vanishes (Brown, 1972). On the other hand, unstable stratification generates waves having a higher critical wavenumber and generates shorter waves, compared to the neutral stratified case. Unstable conditions also affect the orientation of the rolls, arranging them in a more longitudinal direction (Brown, 1972, 1980).
The growth rate of the waves are proportional to the shear at the inflection point. This is determined by how the wind is veering, combined with the change in wind speed with height in the vicinity of the inflection point (Brown, 1980). The maximum growth rate is found when the wave is the most unstable. In order to investigate when and where this maxima would occur, the turning angle, , has been proven to be useful and is defined as the angle between the axis of the BLR and the geostrophic wind, see figure 1. Brown (1972, 1980) has investigated this angle at various times and concludes that during neutral conditions the optimal angle for a maximum growth rate would be =14◦ to the left of the geostrophic wind, but have also reported higher angles in the order of =20-30◦.
2. Method and model setup
2.1 Method
The goal of this study is to analyse the causing mechanisms of boundary layer rolls on four specific dates. The model used in this study perform a linear stability analysis, which inves- tigates how small scale perturbations evolves with time. These perturbations may arise from either the dynamical instability, thermal instability or a combination of the two. By simulating and analyse these instabilities, the mechanisms causing the BLR to arise may be determined.
The results of the linear stability analysis simulations are compared towards data, given by a study of Svensson et al. (2017), where BLR have been observed over the eastern parts of Swe- den, extending out over the island of Gotland, during four days of observations; May 2 & 3, 1997 and May 17 & 25, 2011. The measurements consist of data of the horizontal wind speed components and the potential temperature. The data was obtained from SMHI (Swedish Mete- orological Institute) ground stations. In addition, on May 2 and 3, 1997 a flight observation was conducted over the Baltic Sea, measuring the temperature and wind speed. For further details of how the data was achieved, see section 3. The potential temperature and wind speed may both fluctuate rapidly with time in the atmosphere. In order to be able to analyse these fluctuations,
the perturbation method is used. Perturbations are added into the variables in the equation of motions and the measured data variables can be separated into an average part and a fluctuating part. This will form a set of equations, arranged in such a way that one term describes the per- turbation in the vertical wind field and one term the perturbation in the buoyancy field. These equations form an eigenvalue problem, where the general solution can be represented as a wave having the form of eq (3). Where φ0 is the 2-dimensional perturbation in the buoyancy field or the vertical wind field and ˆφ is a complex. z represents the eigenfunction, σ is the complex growth rate, k and l are the wavenumbers in the zonal and meridional directions respectively.
φ0(x, z, t) = ˆφ(z)e[σt+i(kx+ly)] (3) This wavy motion, with regions of rising and sinking air causes rolls to emerge. As mentioned in section 1.1 and illustrated in figure 1, clouds form in the updraft regions and develop into elongated rolls perpendicular to the wave. In this form, the general solution consists of two types of waves, a kinematic wave, called vorticity wave and an internal gravity wave. The kinematic wave is caused by the perturbation in the vertical wind term (w’), and relates to the dynamic inflection point instability. The internal gravity wave arises due to perturbations in the buoyancy term (b’) and is linked to the thermal instability mechanism (Carpenter et al., 2011).
The general solution gives information about the growth rate of the wave and the phase speed (Holton, 2004). The model also calculates the wavelength, amplitude, the height at which the wave form and the direction of the wave. The wave with the highest growth rate represent the most unstable mode, this wave is the most likely to occur in the atmosphere (Arnqvist et al., 2016). By analysing the above factors, it is possible to establish what instability is causing the BLR to develop.
2.2 Simulation setup
The model performs a linear stability analysis and numerically solves the set of equations de- fined in eq. (4). The full derivation of this equation can be found in section 6.
σ ∇2 0
0 I
! wˆ ˆb
!
=
−¯uik∇2+ ¯ui∇2zk + ν ¯ui∇4z −(k2+ l2)
−∂z∂¯b −ik ¯u + κ∇2z
ˆ w ˆb
!
(4)
In eq. (4) σ=∂t∂ is the growth rate. ∇2 = ∂z∂22 and ∇4z = ∂z∂44. I is the identity matrix, w0 is the perturbations in the vertical wind field, b0-the perturbations in the buoyancy field, ¯u is the mean u-component of the wind. ν is the the kinematic viscosity coefficient. k and l are the wavenumbers in the zonal and meridional directions respectively. κ is the mass diffusivity
(turbulens).
In the simulations setup, a number of input parameters has to be defined. The viscosity (ν) was set to zero. For the turbulence or mass diffusivity (κ), two different setups was used. One where the turbulence was included and estimated as an average turbulent exchange coefficient, calculated as eq. (5). K=0.4 is von Karmans constant and ztop was set to 1500 m. ¯u and ¯v are the mean u- and v-component of the wind respectively.
κ = K2· ¯u
¯ v
max
· ztop
4 (5)
In the other mode, the turbulence was neglected and set to zero. For the potential temperature and horizontal wind speed components, existing data provided by Svensson et al. (2017) was used. The boundary conditions at the bottom of the model was defined as rigid in the velocity field and frictionless for the buoyancy field. At the model top, an insulating boundary condition was chosen for the velocity field and fixed boundary conditions for the buoyancy field. In order to solve the equations in eq. (4), the zonal (k) and meridional (l) wavenumbers have to be defined. These may be set to any arbitrary number. In the simulations, these were both chosen to 50. All combinations of the wavenumbers correspond to a solution, that can be evaluated as a wave. By analysing the growth rate for all combinations of wavenumbers, the wave with the highest growth rate can be identified, which corresponds to the most unstable wave. When the value of the wavenumbers (k) and (l) of the most unstable wave has been determined, the amplitude, height and wavelength (λ) of that wave can be calculated, as defined in equation (6).
λ = 2π
(k2+ l2) (6)
2.3 Analysis approach
Several simulations was carried out, where different simulation setups were used. Two simula- tions tested the dynamic instability, one with turbulence included and defined as in eq. (5), and one without the turbulence, κ=0. In the dynamic simulations, neutral conditions are assumed so the buoyancy-term is neglected, hence -∂¯∂zb=0 becomes in eq. (4). The thermal instability was simulated, both with and without the turbulence. One simulation was conducted including the combination of the thermal and dynamic instabilities, in this simulation the turbulence was set to zero.
The same approach were used in all simulations. First, the coordinate system was turned in steps of 10◦from 0-180◦. For each step, the growth rate of the waves was analysed in search of the one with the highest growth rate. The model only considers the wind profile in a 2D sense, thus the wind will only be affected by shear that acts perpendicular to the u- and v-components.
In order to be able to identify all possible solutions, the wind profile must be turned and analysed over the range of 0-180◦. It is not necessary to investigate the remaining angles (190-360◦) since
they form a symmetric set (Brown, 1980). The coordinate system is defined as meteorological standard. The v- and u-wind components are perpendicular to each other and v is positive in the northward direction and u is positive in the eastward direction. 0◦ is directed at north, 90◦ to the east, 180◦is south and 270◦is west.
When the most unstable wave has been identified, the amplitude, wavelength and height of this wave was analysed and checked to match the observations. Also, the wind direction and orientation of the rolls that arise were evaluated and compared to observations. The buoyancy profile was analysed and the wind profile searched for inflections points. To be able to identify any inflection points, the wind profile has to be rotated and aligned with the direction of the rolls. The rotated wind profile, ˜U , is calculated by eq. (7) as defined in Smyth et al. (2010).
Where k & l are the wavenumbers corresponding to the most unstable wave, u & v are the measured horizontal wind components.
U =˜ ku + lv
√k2+ l2 (7)
If an inflection point exists, the second derivative of the wind (U”) changes sign, which can be found in the wind profile, ˜U , at the point where the curve changes from being concave to convex.
By analysing the solutions corresponding to the w’- and b’-vectors respectively or combined, and compare the results to measurements presented by Svensson et al. (2017), the mechanism causing the rolls can be determined and a conclusion can be drawn.
3. Weather conditions and data
The weather information and data presented in this section are based on the study by Svensson et al. (2017) and further details may be found in their work.
During the following dates; May 2, 1997, May 3, 1997, May 17, 2011 and May 25 2011, BLR was observed over the south-easterly parts of Sweden. SAR- and satellite images confirm that the rolls were visible over the mainland, extending out over the Baltic Sea and the island of Gotland during these days.
Measurements of the daily maximum temperature and the wind speed, at 10 m height, was re- trieved from ground stations operated by SMHI (Swedish Meteorological Institute). In addition, data was also collected during a flight survey over the Baltic Sea on May 2 and 3, 1997. During approximately 1 hour flight, at a height of 30-40 m, about 20 km off the coastline, temperature and wind speed were measured with an accuracy of ±0.25 C◦ and ±0.5 m/s respectively. More details may be found in Svensson et al. (2017).
According to Svensson et al. (2017), the rolls were visible essentially throughout the whole day (0800-1800), except on May 17, 2011, when the BLR only were observed in the afternoon (1300-1900). On May 3, 1997 and May 25, 2011 the rolls formed early in the morning and were
more distinct, with the most prominent rolls over land. While the rolls were less pronounced on May 2, 1997 and May 17, 2011.
During all four days the wind was westerly-northwesterly and the rolls were more or less aligned with the mean wind direction. The weather conditions during all four days showed stable stratification over sea and slightly unstable over land. On May 2 and 3, 1997, the weather conditions were influenced by a low pressure system coming from the Norwegian Sea, moving towards the northern parts of Sweden. This low pressure caused the passage of a cold front on May 2 (Svensson et al., 2017). The wind was directed from (260-300◦) on May 2 and from (290- 320◦) on May 3. The wind speed was 5-10 m/s both days. The maximum temperature during May 2, as measured by SMHI (Swedish Meteorological Institute), was 15-19◦over the mainland and 4-5◦ over sea, giving a temperature gradient of 10-15◦. The same sea surface temperature (4-5◦) was found on May 3, with a temperature over land of 10-14◦. The temperature difference between land and sea was 5-10◦.
There was a front passing in the afternoon on May 17, 2011. The wind speed was 2-12 m/s, south-easterly in the morning and turned to westerly after the passage of the front. The temperature over land was 10-17◦ and 7-10◦over the sea with a temperature difference of 3-6◦. A low pressure system was located over the northern parts of Sweden on May 25, 2011 which gave rise to westerly winds, with wind speeds of 3-9 m/s. The temperature was 10-17◦ over land and 7-10◦over sea, with a temperature gradient of 3-6◦.
Svensson et al. (2017) used the WRF (Weather Research and Forecasting) model to simulate the BLR. They concluded that the rolls form over land, where convection is taking place and are advected out over sea. They found that the amplitude becomes lower and the wavelength shorter as the rolls are transported out over sea on May 2 and 3, 1997. Whereas, on May 17 and 25, 2011 the amplitude and the wavelength are found to be constant over both the mainland and the sea. The results for the wavelength (λ), wind speed, wind direction, amplitude (A) and roll duration, as modelled in WRF by Svensson et al. (2017) for the four dates are presented in table 1.
Table 1. Roll characteristics simulated in WRF by Svensson et al. (2017).
May 2, 1997 May 3, 1997 May 17, 2011 May 25, 2011 Observed rolls (UTC) 0800-1700 0800-1800 1300-1900 0700-1900
Wind speed (ms−1) 15.2 15.3 15.5 17.1
Wind direction (◦) 260-300 290-320 270 270
λland (km) 7.5 7.0 7.8 5.1
λsea(km) 9.9 10.2 9.4 8.7
Aland(ms−1) 1.5 1.7 1.6 1.9
Asea(ms−1) 0.7 1.0 0.9 1.1
4. Results
The neutral, dynamic instability simulations provide interesting results that qualitatively agree with the findings by Svensson et al. (2017), as well as the observations. Therefore, further details of the dynamic studies are presented in the following sections. For convenience, the results are organized by date. This gives a thorough overview of the results and simplifies the analysis of the type of instability mechanism that cause the rolls on each specific date. The results of the study are summarised in table 2.
Table 2. Summarised results from the analysis, showing the characteristics of the most unstable wave. The values in the parenthesis show the observed values found by Svensson et al. (2017). The notation ”-” indicates that no results that agree with the observations can be found in the analysis.
May 2, 1997 May 3, 1997 May 17, 2011 May 25, 2011 Wind direction(land)(◦) 280-300 300-320 255-270 270-295
Wind direction(sea)(◦) 275-300 300-320 230-260 260-290 Observed wind direction (◦) (260-300) (290-320) (270) (270)
Mean wind direction(land)(◦) 291 307 263 274
Mean wind direction(sea) (◦) 288 307 251 269
Direction of rolls(land)(◦) 288 (260-300) 307 (290-320) 257 (270) 295 (270) Direction of rolls(sea) (◦) - 307 (290-320) - 285 (270) λland (m) 4069 (7500) 3347 (7000) 4667 (7800) 1667 ( 5100)
λsea(m) - 1343 (10 200) - 2383 (8700)
Aland(ms−1) 0.74 (1.5) 0.71 (1.7) 0.89 (1.6) 0.83 (1.9)
Asea(ms−1) - 0.75 (1.0) - 0.72 (1.1)
Height to the rolls(land)(m) 606 621 1000 984
Height to the rolls(sea)(m) - 818 - 606
Growth rate(land)(s−1) 1.05·10−4 1.45·10−4 1.96·10−4 2.05·10−4
Growth rate(sea) (s−1) - 7.15·10−5 - 4.18·10−4
4.1 May 2, 1997
4.1.1 Land
The most unstable wave occurs for the neutral, dynamic instability study. The wave is oriented at 198◦, with rolls forming perpendicular to the wave, at a direction from 288◦. On May 2, 1997 the wind is coming from north-west (280-300◦), with a mean wind direction of 291◦. Figure 4 a) shows the measured data for the wind directions throughout the day. Since the rolls are aligned with the mean wind direction, figure 4 a) also shows the orientation of the observed rolls. Figure 4 b) shows the direction of the simulated wave (blue) and rolls (red).
The direction of the simulated rolls agree well with the observations, figures 4 a and b). The modelled wavelength is 4069 m, the growth rate of the wave is found to be 1.05·10−4 s−1.
Figure 4 c) shows the growth rate, where the most unstable wave is indicated by the black line.
The height of the wave is 606 m with an amplitude of 0.74 ms−1, shown in figure 5. When analysing the wind profile, figure 5, an inflection point can be seen around the height of ∼ 600 m.
(a) (b)
(c)
Figure 4. Results of the neutral, dynamic simulation over land on May 2, 1997. a) The observed directions from which the wind is coming throughout the day. The blue arrows represent the wind speed. The circles show the wind speed, with an increment of 5 ms−1 for each circle. b) The direction of the simulated wave (blue) and rolls (red). c) The growth rate of the most unstable wave. The black line indicate the simulated wave. The x- and y-axes show the zonal (k) & meridional (l) wave numbers respectively and the colour-bar shows the growth rate of the most unstable wave [s−1].
(a) (b)
Figure 5. Results for the dynamic instability over land on May 2, 1997. a) The vertical wind profile aligned with the direction on the rolls. b) The amplitude and height of the most unstable wave. Shown on the x-axis is the amplitude and the height of the wave on the y-axis.
4.1.2 Sea
On May 2, 1997, the wind direction is northwesterly, 275-300◦, with a mean wind direction of 288◦, figure 6. No results that agree with the observations can be produced over sea on May 2, 1997. The dynamic instability study results in a wavelength much shorter (∼ 1000 m), than the observations (∼ 9000 m) and no distinct inflection point can be found in the wind profile. When the thermal instability is examined, the most unstable wave appears at a very low height (< 45 m), which is inconsistent with the observations. The same results are found whether turbulence is included or not. The study of the combination of the thermal and dynamic instabilities results in a wavelength much shorter (λ< 214 m) than the one being observed.
Figure 6. The wind direction over sea during May 2, 1997. The blue arrows represent the wind speed. The circles show the wind speed, with an increment of 5 ms−1 for each circle.
4.2 May 3, 1997
4.2.1 Land
On May 3, 1997 the wind over land is coming from north-west, 300-320◦, with a mean wind direction of 307◦. Figure 7 a) shows the measured wind direction during the day and also the orientation of the observed rolls. The neutral, dynamic instability simulation generates the most unstable wave. The wave appears oriented at 217◦, with the rolls forming in a direction from 307◦, shown in figure 7 b). The observed rolls match the direction of the rolls generated in the simulation. The growth rate is 1.45·10−4 s−1, figure 7 c). The wavelength is 3347 m, the wave has an amplitude of 0.71 ms−1 and occur at a height of 621 m, shown in figure 8. This height coincides with the inflection point, which can be seen at approximately ∼ 600 m height in figure 8.
(a) (b)
(c)
Figure 7. a) The measured wind direction during the day. The blue arrows represent the wind speed. The circles show the wind speed, with an increment of 5 ms−1 for each circle. b) The direction of the wave (blue) and the rolls (red) simulated with the dynamic case without turbulence. c) The growth rate for the dynamic simulation.
The black line indicates the direction and wave number magnitude of the most unstable wave.
(a) (b)
Figure 8. Results of the dynamic simulation without turbulence over land on May 3, 1997. a) The wind profile. b) The height and amplitude of the most unstable wave.
4.2.2 Sea
The most unstable wave is found when studying the dynamic instability, under neutral stratifi- cation and without turbulence. The mean wind direction is 307◦ and the wind is coming from north-west, 300-320◦. Figure 9 a) shows the wind direction and orientation of the observed rolls on May 3, 1997. Figure 9 b) shows the direction of the simulated wave (217◦) and rolls (307◦) and it can be seen that the direction of the simulated rolls are in good agreement with the orientations of the observed rolls. The growth rate is 7.15·10−5 s−1shown in figure 9 c), where the most unstable wave is indicated by the black line. The wave arise at a height of 818 m, with an amplitude of 0.75 ms−1, seen in figure 10. There is an inflection point at ∼ 800 m height, seen in figure 10, this figure also shows the height of the wave. However, the wavelength is 1343 m, which is much shorter compared to the observed wavelength of the rolls (10 200 m).
(a) (b)
(c)
Figure 9. a) The measured direction of the wind during May 3, 1997. The blue arrows represent the wind speed.
The circles show the wind speed, with an increment of 5 ms−1 for each circle. b) The simulated direction of the most unstable wave (blue) and rolls (red) for the dynamic case. c) The grow rate, where the black line indicates the direction and wave number magnitude of the most unstable wave. The x- and y-axes show the zonal (k) &
meridional (l) wave numbers respectively and the colour-bar to the right shows the growth rate [s−1].
(a) (b)
Figure 10. Results of the dynamic simulation. a) Shows the wind profile. b) The height and amplitude of the most unstable wave.
4.3 May 17, 2011
4.3.1 Land
Both the simulation of the dynamic instability where turbulence is included as well as without the turbulence generate results in agreement with observations for May 17, 2011. The observed rolls are aligned approximately with the wind over land on May 17 2011, which is mainly westerly, 255-270◦ and shown in figure 11 a). In figure 11 b) it can be seen that the simulated rolls are oriented from 257◦, and compares well to the observed direction of the rolls, which are aligned with the wind direction. The growth rate (1.96·10−4s−1) and the most unstable wave (black line) is shown in figure 11 c). The height of the most unstable wave (1000 m), the wavelength (4667 m), growth rate (1.96·10−4s−1) and direction of the rolls (257◦) are the same regardless if turbulence is included or not. The only parameter that differs is the amplitude, which is 0.71 ms−1 without the turbulence and 0.89 ms−1 with turbulence included. When compared to the observed amplitude of the wave (1.6 ms−1), the study including the turbulence generates an amplitude (0.89 ms−1) more in agreement to the observations. Therefore, this simulation is considered to match the observations to a higher degree. Figure 12 shows the wind profile, the amplitude and height of the wave. An inflection point can be distinguished at approximately 1000 m height, in agreement with the height at which the rolls form.
(a) (b)
(c)
Figure 11. a) The measured wind direction on May 17, 2011 over land. The blue arrows represent the wind speed.
The circles show the wind speed, with an increment of 5 ms−1 for each circle. b) The direction of the simulated wave (blue) and rolls (red) for the dynamic case, under the influence of turbulence. c) The growth rate of the most unstable wave. The black line indicates the direction and wave number magnitude of the most unstable wave.
(a) (b)
Figure 12. a) The wind profile, aligned with the direction of the rolls on May 17, 2011. b) The height and amplitude of the most unstable wave for the dynamic study including the turbulence.
4.3.2 Sea
On May 17, 2011 the wind is southwesterly, 230-260◦ with a mean wind direction of 251◦, figure 13. No results that agree with the observations can be produced over sea for any of the studied instability cases. The dynamic instability study shows that the wave and the rolls arise at a height that is too low, 75 m or less, compared to the observations. When the thermal instability is examined, the most unstable wave appears at a very low height (< 45 m), which is inconsistent with the observations. The same results are found whether turbulence is included or not. In the study of the combination of the thermal and dynamic instabilities the wavelength is found to be much shorter (λ< 214 m) than the observed wavelength.
Figure 13. The observed wind direction throughout the day over sea on May 17, 2011.The blue arrows represent the wind speed. The circles show the wind speed, with an increment of 5 ms−1 for each circle.
4.4 May 25, 2011
4.4.1 Land
The most unstable wave is found for the neutral, dynamic instability study. The wave is directed at 205◦and the rolls form oriented from 295◦, figure 14 a). On May 25, 2011 the wind is west- erly (270-295◦) and the rolls form aligned with the wind direction. The observed orientation of the rolls agree with the direction of the rolls in the simulation. Figure 14 b) shown the wind direction and orientation of the observed rolls. The most unstable wave has a growth rate of 2.05·10−4s−1, figure 14 c). The wavelength is 1667 m, which is shorter than what is found in measurements (5100 m). The wave has an amplitude of 0.83 ms−1 and is found at a height of 984 m, seen in figure 17, also showing the wind profile and an inflection point can be seen at ∼ 1000 m height.
(a) (b)
(c)
Figure 14. a) The wind direction over land throughout the day May 25 2011. The blue arrows represent the wind speed. The circles show the wind speed, with an increment of 5 ms−1 for each circle. b) The direction of the simulated wave (blue) and roll (red) over land May 25, 2011. c) The growth rate of the most unstable wave,indicated in black.
(a) (b)
Figure 15. a) The wind profile, aligned with the direction of the rolls on May 25, 2011. b) The height and amplitude of the most unstable wave for the dynamic simulation.
4.4.2 Sea
The dynamic instability simulation without the turbulence generates the most unstable wave.
The orientation of the wave is 195◦and the rolls are directed from 285◦, figure 16 a). The wind direction over sea on May 25, 2011 is westerly to north-westerly (260-290◦) with a mean wind direction of 269◦, and the observed rolls form aligned with the wind direction, shown in figure 16 b). The growth rate is 4.18·10−4s−1, figure 16 c). The wave appear at a height of 606 m, with an amplitude of 0.72 ms−1, figure 17. In this figure, an inflection point can be distinguished at the height of ∼ 600 m. The wavelength found in the simulations (2383 m) is shorter than found by Svensson et al. (2017) (8700 m).
(a) (b)
(c)
Figure 16. a) The measured wind direction over sea throughout May 25, 2011. The blue arrows represent the wind speed. The circles show the wind speed, with an increment of 5 ms−1 for each circle. b) The direction of the wave (blue) and the orientation of the rolls (red) for the dynamic simulation, without turbulence. c) The grow rate of the most unstable wave. The black line indicates the direction and wave number magnitude of the most unstable wave.
The x- and y-axes show the zonal (k) & meridional (l) wave numbers respectively and the colour-bar to the right shows the growth rate [s−1].
(a) (b)
Figure 17. a) The wind profile, aligned with the direction of the rolls on May 25, 2011. b) The height and amplitude of the most unstable wave for the dynamic simulation.
(a) (b)
(c) (d)
Figure 18. The buoyancy profile over land for a) May 2, 1997. b) May 3, 1997. c) May 17, 2011. d) May 25, 2011
5. Discussion
The linear stability analysis indicates that the most unstable wave arises due to the dynamic instability. Rolls that agree with the observations can be modelled over land on all four days;
May 2, 1997, May 3, 1997, May 17, 2011 and May 25, 2011 and over sea surface on May 3, 1997 and May 25, 2011. The rolls are visible over both land and sea surface on all studied days and the wind direction is from west to north-west. This excludes the rolls forming over sea, because if the rolls were to form over sea surface only, they would not be observed over land.
The analysis indicates that the rolls form over the mainland and are advected out over the sea.
However, the rolls might be maintained by the dynamic instability over sea on May 3, 1997 and May 25, 2011, even if the rolls initially form over land on these dates.
The modelled wavelength on May 3, 1997 & May 25, 2011 is in the range of λ∼ 1300-2400 m over sea and λ∼ 1600-2900 m over land. These results are somewhat lower than presented by Svensson et al. (2017), where wavelengths are in the range of 8700-10 200 m over the sea and 5100-7800 m over the mainland. Even though the wavelengths found in this study deviate from the wavelengths presented by Svensson et al. (2017), it still agrees well with the characteristic wavelength found in other studies. Kuettner (1971) finds an average wavelength of 2-8 km, which compares well with the results found in this study.
The results are essentially the same whether or not turbulence is included in the dynamic instability study or not, except over sea on May 3, 1997. In this case, no results that agree with the observations can be found if turbulence is included into the dynamic model. This might be explained by how the turbulence is defined (eq. 5). The calculated value of the constant turbulent coefficient is probably too low to influence the dynamic instability significantly.
The combination of the thermal and dynamic instabilities generate inconsistent results. In this case, the wavelength becomes <149 m and the height <45 m over both land and sea surface.
The thermal instability does not seem to contribute to the formation of the rolls. Although rolls are generated in the thermal instability simulations, the most unstable wave either arises very close to the ground, at about ∼15-45 m height, or has an amplitude close to zero. These results that are inconsistent with findings by Svensson et al. (2017). These findings are expected over sea surface, where stable conditions prevails and hence there is no convection over sea that would cause any thermal instabilities to arise. Over land, the atmospheric conditions are unstable and convection may occur up to a height of about 500-550 m. Above ∼ 550 m the air is neutral or stable as seen in figure 18. According to theory, convection in the presence of a curved wind profile triggers the formation of BLR (Kuettner, 1971). As seen in the figures of the wind profile over land the four days; figures (5, 8, 12, 17) the wind speed is increasing the most in the lowest ∼100 m above ground and the wind profile is the most curved at a relatively low level. The thermal instability is therefore most pronounced close to the ground, which explains why the most unstable wave occurs at this height. The altitude of the most unstable wave and thus also the rolls in the simulations are inconsistent with the observations, and so
the thermal instability mechanism is not considered to contribute to the formation of the rolls on the studied days. None of the studied instabilities can produce results in agreement with the observations over sea on May 2, 1997 and May 17, 2011. The dynamic instability analysis results in wavelengths that are much shorter (λ ∼ 1000 m) than what is observed (λ ∼ 10 000 m) and no inflection point can be identified in the wind profile for these days. These findings oppose the dynamic instability as the cause of the BLR over sea on May 2, 1997 and May 17, 2011. The lack of results over sea on May 2, 1997 and May 17, 2011, is probably due to the fact that the observed rolls are weaker and less pronounced these days, as found by Svensson et al. (2017).
Svensson et al. (2017) found that the rolls appear at a lower altitude over sea surface than over land. They also present results indicating that the amplitude of the rolls and the wavelength are lower over sea compared to over land on May 2 & 3, 1997, whereas the amplitude and wavelength are constant over sea as well as over land on May 17 & 25, 2011. The outcome of this study show inconclusive results regarding these features. The rolls do arise at a lower altitude over sea compared to over land and the wavelength is shorter over sea on May 3, 1997.
On the contrary, the amplitude is slightly higher over sea than over land May 3, 1997. The results of May 25, 2011 are inconsistent with the findings by Svensson et al. (2017). The amplitude is more or less constant over sea and over land, however the rolls form at a higher altitude over land than over sea. Also the wavelength is higher over sea compared to land. The lack of results over sea on May 2, 1997 and May 17, 2011 make it impossible to draw any conclusions on these dates.
6. Conclusions
This study examines the origin of boundary layer rolls for a specific case study by Svensson et al. (2017). The forcing mechanisms, causing rolls to emerge over the Baltic Sea during four days in May 1997 & 2011, have been analyzed in this study. By performing a linear stability analysis, the dynamic instability is found to be the main mechanism causing the boundary rolls on the following dates: May 2, 1997, May 3, 1997, May 17, 2011 and May 25, 2011. Svensson et al. (2017) concludes that the rolls form over the mainland in a convective airmass and are advected over the sea surface.
The rolls can be modelled over land on all four studied dates, as well as over sea surface on May 3, 1997 and May 25, 2011. No rolls can be produced in any of the simulations over sea on May 2, 1997 & May 17, 2011. Based on these results, the conclusion is drawn that the rolls form primarily over land and are advected out over the sea, in accordance with the findings by Svensson et al. (2017). The results also indicate that the rolls might be maintained by the dynamic instability over sea on May 3, 1997 and May 25, 2011, even if the rolls initially form over land these dates.
The simulated wavelength, amplitude and height to the rolls all qualitatively agree with the
observations presented in Svensson et al. (2017), although slightly lower values are found in this study. Also, an inflection point can be seen at approximately the same height as the most unstable wave appears in the wind profiles. These findings support the dynamic instability as the causing mechanisms of the rolls these days. The wind direction and orientation of the rolls, found over both land and sea surface in this study are consistent with the observations presented in Svensson et al. (2017).
References
Arnqvist, Johan. Bergström, Hans. Nappo, Carmen. (2016). Examination of the mechanism behind observed canopy waves. Agricultural and Forest Meteorology: Vol. 218-219 pp. 196- 203.
Brown, R, A. (1970). A Secondary Flow Model for the Planetary Boundary Layer. Journal of the atmospheric sciences: Vol. 27, pp. 742-757.
Brown, R, A. (1972). On the Inflection Point Instability of a Stratified Ekman Boundary Layer.
Journal of the atmospheric sciences: Vol. 29, pp. 850-859.
Brown, R, A. (1980). Longitudinal Instabilities and Secondary Flows in the Planetary Boundary Layer: A Review. Reviews of Geophysics and Space physics: Vol. 18, pp. 683-697.
Brown, R, A. (1980). On the use of exchange coefficients in modelling turbulent flow.
Boundary-Layer Meteorology: Vol. 20, pp. 111-116.
Carpenter, Jeffrey, R. Balmforth, N, J. Lawrence, Gregory, A. (2010). Identifying unstable modes in stratified shear layers. Physics of fluids: Vol. 22, 054104.
Carpenter, Jeffrey, R. Tedford, Edmund, W. Heifetz, Eyal. Lawrence, Gregory, A. (2011). Insta- bility in Stratified Shear Flow: Review of a Physical Interpretation Based on Interacting Waves.
Applied Mechanics Reviews, Vol.64, /060801-1.
Christian, Thomas, W. Wakimoto, Roger, M. (1989). The relationship between radar reflectivi- ties and clouds associated with horizontal roll convection on 8 August 1982. Monthly Weather Review, 07/1989, Vol. 117, Number 7.
Etling, D. Brown, R, A. (1992). Roll vortices in the planetary boundary layer: A review.
Boundary-Layer Meteorology: Vol. 65, pp. 215-248.
Grossman, Robert, L. (1982). An analysis of vertical velocity spectra obtained in the bomex fair-weather, trade-wind boundary layer. Boundary Layer Meteorology: Vol. 23, pp. 323-357.
Holton, James. R. (2004). An introduction to dynamic meteorology, Fourth Edition. Elsevier Academic Press, 2004. ISBN: 0-12-354015-1.
Kuettner, Joachim, P. (1965). Cloudstreets, theory and satellite observations, 10 OSTIV Congress, South Cerney (England), June 1965.
Kuettner, Joachim, P. (1971). Cloud band in the earth’s atmosphere: Observations and Theory.
TellusVol. 23, Issue 4-5, pp. 404-426.
LeMone, Margaret, Anne. (1973). The Structure and Dynamics of Horizontal Roll Vortices in the Planetary Boundary Layer. Journal of the atmospheric sciences Vol. 30, pp.1077-1091.
Markowski, Paul M. Richardson, Yvette P. (2010). Mesoscale Meteorology in Midlatitudes, Advancing Weather and Climate Science Ser. Second edition, John Wiley Sons, Incorporated 2010. Chapter 3, pp: 41-71.
Müller, Sabine. Stanev, Emil, V. Schulz-Stellenfleth, Johannes. Staneva, Joanna. Koch, Wolf- gang. (2013). Atmospheric boundary layer rolls: Quantification of their effect on the hydrody- namics in the German Bight. JOURNAL OF GEOPHYSICAL RESEARCH: OCEANS, Vol. 118, 5036-5053, doi:10.1002/jgrc.20388.
Rabinovich, A. Umurhan, O, M. Harnik, N. Lott, F. Heifetz, E. (2011). Vorticity inversion and action-at-a-distance instability in stably stratified shear flow. J. Fluid Mech. Cambridge University Press (2011), Vol. 670, pp. 301-325.
Sandeepan, B,S. Rakesh, P,T. Venkatesan, R. (2013). Observation and simulation of boundary layer coherent roll structures and their effect on pollution dispersion. Atmospheric Research Vol. 120-121, pp. 181-191.
Smyth, W, D. Moum, J, N. Nash, J, D. (2010). Narrowband oscillations in the upper equatorial ocean. Part II: Properties of shear instabilities. JOURNAL OF PHYSICAL OCEANOGRAPHY:
Vol. 41, pp. 412-428.
Svensson, Nina. Saleé, Erik. Bergström, Hans. Nilsson, Erik. Badger, Merete. Rutgersson, Anna. (2017). A Case Study of Offshore Advection of Boundary Layer Rolls over a Stably Stratified Sea Surface. Advances in Meteorology: Vol. 2017, Article ID 9015891, 15 pages.
Svensson, Nina. (2018). Mesoscale Processes over the Baltic Sea. ISBN 978-91-513-0331-4.
Weckwerth, Tammy, M. Wilson, James, W. Wakimoto, Roger, M. (1996). Thermodynamic Variability within the Convective Boundary Layer Due to Horizontal Convective Rolls. Monthly Weather Review: Vol. 124, pp. 769-784.
Weckwerth, Tammy, M. Wilson, James, W. Wakimoto, Roger, M. Crook, Andrew, N. (1997).
Horizontal Convective Rolls: Determining the Environmental Conditions Supporting their Ex- istence and Characteristics. Monthly Weather Review: Vol. 125, pp. 505-526.
Weckwerth, Tammy, M. Horst, Thomas, W. Wilson, James, W. (1998). An Observational Study of the Evolution of Horizontal Convective Rolls. Monthly Weather Review: Vol. 127, pp. 2160- 2179.
Appendix: Basic equations
6.1 Definition of parameters
σ - growth rate [s−1] ν - viscosity [m2s−1] κ - mass diffusivity (turbulens) [m2s−1] B - buoyancy profile
k - wavenumber in x-direction N - Brunt Väisälä frequency [s−1] l- wavenumber in y-direction λ - wavelength [m]
w’ - vertical wind eigenfunction b’ - buoyancy eigenfunction u - zonal wind [ms−1] v - meridional wind [ms−1]
6.2 Derivation of the equations
The goal of this section is to derive an expression describing the perturbations in the vertical wind field (w’) and the buoyancy field (b’). These are used in the linear stability analysis when the mechanisms causing the BLR are analysed. The derivations are based on and follow the scheme presented by Smyth et al. (2010). The starting point of the derivations are the equations of motion eq. (8), the continuity equation eq. (9) and the thermodynamic equation eq. (10).
The derivation consists of two steps, one in which an expression for the vertical perturbation (w’) is derived by manipulating the momentum equations. In the second step, the perturbations in the buoyancy field is derived from the thermodynamic equation.
Du Dt = −1
ρ
∂p
∂x + f v + Frx Dv
Dt = −1 ρ
∂p
∂y − f u + Fry Dw
Dt = −1 ρ
∂p
∂z + g θ θ0 + Frz
(8)
∂u
∂x + ∂v
∂y +∂w
∂z + ω d
dz(lnρ0) = 0 (9)
Db
Dt = κ∂2b
∂z2 (10)
In the equations above, ρ is the density, p is the pressure, f is the Coriolis parameter, u & v are the horizontal components of the wind, w is the vertical wind component, g is the gravitational constant. θ is the fluctuations of the potential temperature compared to a standard value, θ0. κ is the mass diffusivity (turbulens), b- the buoyancy, Frx, Fry and Frz are the frictional forces in x, y, z directions respectively, ν is the kinematic viscosity coefficient. Equation (11) relates the potential temperature to the buoyancy force. The frictional forces are defined as in equation
(12).
g θ
θ0 = gρ0− ρ
ρ = b (11)
Frx = ν ∂2u
∂x2 +∂2u
∂y2 + ∂2u
∂z2
Fry = ν ∂2v
∂x2 +∂2v
∂y2 + ∂2v
∂z2
Frz = ν ∂2w
∂x2 +∂2w
∂y2 + ∂2w
∂z2
(12)
Step 1: We begin by manipulating the horizontal momentum equations, (8). Since we are deal- ing with small horizontal scales in the momentum equation, the rotation of the earth does not affect the solutions. Hence, we can make the following assumptions; the Coriolis force can be neglected and there is no planetary rotation Holton (2004). The Boussinesq approximation is valid in the horizontal, i.e ρ is constant and the density-terms can be neglected in the horizontal momentum equations. However, this is not the case in the vertical direction, which is why we keep the buoyancy term in the vertical momentum equation. By applying the above assump- tions, replace the buoyancy term in the vertical momentum equation with the definition in eq.
(11) and use the expressions for the frictional forces, eq. (12), the momentum equations eq. (8) can be rewritten as eq. (13).
Du
Dt = −∂p
∂x + ν ∂2u
∂x2 + ∂2u
∂y2 +∂2u
∂z2
Dv
Dt = −∂p
∂y + ν ∂2v
∂x2 +∂2v
∂y2 +∂2v
∂z2
Dw
Dt = −∂p
∂z + b + ν ∂2w
∂x2 + ∂2w
∂y2 +∂2w
∂z2
(13)
D Dt = ∂
∂t+ u ∂
∂x + v ∂
∂y + w ∂
∂z = 0 (14)
Next, using the definition of the total time-derivative, eq. (14) on the right-hand side of eq. (13) and rearranging we get eq. (15). Since the horizontal scales are much greater than the vertical scale, the second derivatives in the x- and y-directions on the left-hand side of the horizontal components of eq. (13) are small and can therefore be neglected. Similarly, the second deriva- tives of x and y drop out of the vertical momentum equation as well. The w-component of the wind is much smaller than the v- & u-components and so the second derivative of z vanishes in the vertical equation.
