QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY Q. J. R. Meteorol. Soc. 133: 265–271 (2007) Published online in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/qj.27
Notes and Correspondence Further comments on the equilibrium height of neutral and stable planetary boundary layers Sergej Zilitinkevich,a,b* Igor Esaub and Alexander Baklanovc b
a Division of Atmospheric Sciences, University of Helsinki, Finland Nansen Environmental and Remote Sensing Centre/Bjerknes Centre for Climate Research, Bergen, Norway c Danish Meteorological Institute, Copenhagen, Denmark
ABSTRACT: The height of the neutrally and stably stratified turbulent planetary boundary layers (PBLs) in the atmosphere and hydrosphere is controlled by numerous factors of different natures. The PBL deepening is facilitated by turbulent mixing caused, first of all, by the velocity shear at the surface, and prevented by the Earth’s rotation and the negative buoyancy forces caused by the flow–surface interaction and the stable static stability in the free atmosphere above the PBL. The balance between these factors is often complicated by the baroclinic shears and large-scale vertical motions (usually disregarded in field experiments), not to mention temporal and horizontal variability of natural PBL flow. It is not surprising that until now many alternative (sometimes contradictory) formulations for the stable PBL height have been proposed and no consensus achieved. To the authors’ opinion, the most natural way to investigate this complex problem is to consider step by step the PBL height in comparatively simple, basic equilibrium regimes (controlled by limited factors), to develop and to validate theoretical models for these regimes, and only afterwards to switch to general diagnostic (equilibrium) and prognostic (non-steady) equations. In the present paper, we briefly summarize recent investigations following the above strategy, and present new data supporting our earlier theoretical conclusions and recommendations. We would not recommend using PBL height equations inconsistent with firmly established basic-regime formulations. Copyright 2007 Royal Meteorological Society KEY WORDS
boundary-layer height; stratification; baroclinicity; Earth’s rotation; free-flow stability; relaxation; meteorological measurements; large-eddy simulation
Received 6 April 2006; Revised 15 October 2006; Accepted 15 October 2006
1.
Introduction and discussion
In boundary-layer meteorology, the term ‘neutral’ was traditionally applied to the planetary boundary layer (PBL) with zero buoyancy flux at the surface (Bs = 0), and the term ‘stable’ (or ‘nocturnal’ considered as equivalent to ‘stable’) to the PBL with Bs < 0, regardless of the static stability in the free flow above the PBL. Recently, we have demonstrated that the free-flow stability (characterized by the free-flow Brunt–V¨ais¨al¨a frequency, N ) strongly affects all properties of the PBL including the surface-layer flux-profile relationships (Zilitinkevich and Calanca, 2000; Zilitinkevich, 2002; Esau, 2004b; Zilitinkevich and Esau, 2005) and the PBL height (Baklanov, 2002; Zilitinkevich et al., 2002; Zilitinkevich and Baklanov, 2002; Zilitinkevich and Esau, 2002, 2003). Accounting for the effect of N , we distinguish between the following four (instead of two) types of neutral and stable atmospheric PBLs: * Correspondence to: Sergej Zilitinkevich, Division of Atmospheric Sciences, University of Helsinki, Finland. E-mail:
[email protected] Copyright 2007 Royal Meteorological Society
• ‘Truly neutral’ (TN) with Bs = 0, N = 0, • ‘Conventionally neutral’ (CN) with Bs = 0, N > 0, • ‘Nocturnal stable’ (NS, in other words short-lived) with Bs < 0, N = 0, and • ‘Long-lived stable’ (LS) with Bs < 0, N > 0 (which possess features of the latter two basic regimes). To derive a general formulation for the equilibrium PBL height,√hE , we apply the familiar Ekman (1905) formula, hE ∼ KM /|f |, where f is the Coriolis parameter and KM is the eddy viscosity scale, in combination with scaling estimates of KM in the three basic regimes: u∗ hE for TN PBL K M h2E ∼ , KM ∼ uT lT ∼ u∗ LN for CN PBL |f | u∗ L for NS PBL. (1) Here in all cases the friction velocity, u∗ , is used as the turbulent velocity scale, uT , whereas the turbulent lengthscale, lT , is taken proportional to • the PBL height hE in the TN PBL, • the length scale LN = u∗ N −1 , introduced by Kitaigorodskii and Joffre (1988), in the CN PBL,
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• and the Monin–Obukhov length-scale L = −u3∗ Bs−1 (Monin and Obukhov, 1954) in the NS PBL.
are taken equal to unity: i.e. CR = CCN = CNS = 1. The comments given in section 2 of Steeneveld et al. (2007) are apparently caused by overlooking this fact.
This analysis gives PBL height equations
hE ∼
u∗ |f |−1 u∗ |f N |−1/2
2 −1/2 u∗ |f Bs |
(Rossby and Montgomery, 1935) for TN PBL (Pollard et al ., 1973) for CN PBL (Zilitinkevich, 1972) for NS PBL,
2. Why is PBL height difficult to determine from atmospheric data? (2)
relevant to alternative equilibrium PBL regimes: TN, CN or NS. Rossby and Montgomery (1935) did not distinguish between TN and CN PBLs, and neither did Pollard et al. (1973). They derived the expression u∗ |f N |−1/2 for the maximal depth of the oceanic upper mixed layer deepening against the stably stratified thermocline (with gρ0−1 ∂ρ/∂z = N 2 ) in the course of inertial oscillations. Kitaigorodskii and Joffre (1988) proposed the scale LN = u∗ N −1 as relevant to the PBL height. In our analysis, LN is treated as the length-scale of turbulent eddies in the interior of the CN PBL rather than the PBL height-scale. It is conceivable that in the general case, the dominant role is played by the smallest length-scale (which imposes the strongest limit on lT ). This requirement is satisfied with any interpolation of the type h−n E = −n −n h−n + h + h with n > 0. In earlier work, we E(TN) E(CN) E(NS) have verified different types of interpolation against our large-eddy simulation (LES) database on stably and neutrally stratified PBLs computed with the Nansen Centre LES code LESNIC (Esau, 2004a, b). Results from these LESs are in broad agreement with independent LES studies by Mason and Thompson (1987, 1992). The best fit was achieved at n = 2. It implies linear interpolation between the squared reciprocals of hE : 1 h2E
=
f2 2
(CR u∗ )
+
N |f | 2
(CCN u∗ )
+
|f Bs | (CNS u2∗ )2
,
(3)
where CR , CCN and CNS are dimensionless empirical 1/2 constants. Because hE ∝ KM , Equation (3) also follows from the linear interpolation between reciprocals of KM . Physically essential are the three fundamental PBL height formulations (Equation (2)), whereas the way of interpolation – through Equation (3) or otherwise – represents a practically important but rather technical issue. We recall that, to be consistent, a PBL height formulation should reduce to the limits given by Equation (2) in the TN, CN and NS PBL regimes. Steeneveld et al. (2007) quoted Equation (2) (see their Equations (5)–(7)), but apparently neglected the above requirement. As a result their Equation (14) leads to obviously erroneous conclusions, namely, that the PBL height tends to infinity when either N or Bs tend to zero. Notice also that Equation (10) of Steeneveld et al. (2007), presented as an alternative to our Equation (3), is nothing but our Equation (3) (reproduced as their Equation (9)), with the only difference that all coefficients Copyright 2007 Royal Meteorological Society
Clearly our Equation (3) is one of reasonable interpolations involving three dimensionless empirical coefficients, CR , CCN and CNS , and four governing parameters, u∗ , f, N, Bs ; hence we have two dimensionless stability parameters: µ = u∗ |f L|−1 and µN = N/|f |.
(4)
In the real atmosphere, the equilibrium PBL height depends also on the geostrophic wind shear, = |∂ug /∂z|, which increases hE (Zilitinkevich and Esau, 2003), and on the large-scale vertical velocity, wh , which increases or decreases hE by the increment wh tPBL , where tPBL ∼ hE /u∗ is the PBL relaxation time-scale (Zilitinkevich and Baklanov, 2002). This involves two independent dimensionless parameters, µ = /N and µw = wh /u∗ , usually unavailable in atmospheric boundary-layer experiments. It is not surprising that atmospheric data on the stable/neutral PBL height always show a very wide spread. The situation is additionally complicated when data on (or analysis of) the PBL height are presented without saying how this height is defined. Recall that Equations (1)–(3) are applied to the equilibrium height, hE , of the Ekman boundary layer, that is the turbulent boundary layer in a rotating, stratified fluid. In this context, we apply the commonly accepted definition of the upper boundary of the turbulent boundary layer as the height z = h, at which the vertical turbulent flux of momentum |τ | reduces to 5% of its value at the surface (in other words, |τ |z=h = 0.05 u2∗ ). Unfortunately atmospheric measurements rarely include data on the upper portion of the vertical profile of |τ |. Alternative determinations of the stable PBL height from the mean wind and temperature profiles (e.g. Troen and Mahrt, 1986; Seibert et al., 2000; Piringer and Joffre, 2005) are very uncertain in cases with no or weak temperature inversions at the PBL upper boundary (cf. discussion of the bulk Richardson number approach in Zilitinkevich and Baklanov, 2002). It worth emphasizing that, in current observational practice, the PBL height is not measured ‘directly’, i.e. from data on the vertical profile of |τ | or another relevant parameter characterizing turbulence. Indirect methods, especially as applied to stable PBL height, include theoretical interpretations of measurements and are not perfect (Seibert et al., 2000; Piringer and Joffre, 2005). As reported by Piringer and Joffre (2005), comparisons of various diagnostic formulations with available experimental data (mainly from sodar- and radio-sounding) disclosed large scatter and bias and poor correlation. In Q. J. R. Meteorol. Soc. 133: 265–271 (2007) DOI: 10.1002/qj
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these circumstances, we are forced to admit that the currently available experimental data on the atmospheric PBL height cannot be considered as the ‘empirical truth’. All things considered, direct verification of Equation (3) or any other theoretical formula of this type against atmospheric data does not look promising; even in the ideal case, when all the required parameters are measured, we will need to consider empirical functions of two or more arguments and to determine three (as in Equation (3)) or more empirical constants at once. 3.
New LES data on PBL height
In so complex problem as the stable PBL height, it is reasonable to start by examining particular regimes against carefully selected data with all governing parameters under control. According to the theory, Equation (3) holds true in the steady-state, barotropic PBL at wh = 0. To verify this equation, we obviously need data representing essentially different stratification regimes. As already mentioned, available atmospheric data are hardly suitable for this purpose. Therefore we are forced to limit our analysis to data from LESs. In this paper we employ LES data generated using the Nansen Centre code LESNIC. It solves the momentum, temperature and continuity equations for an incompressible Boussinesq fluid and employs advanced numerical schemes: a fully conservative second-order centraldifference scheme for the skew-symmetric advection terms; a fourth-order Runge–Kutta scheme for time stepping; and a direct fractional-step pressure correction scheme for continuity preservation. The computational mesh is the staggered C-type mesh, which demands turbulent fluxes only as boundary conditions. The code employs dynamic mixed closure, which allows the avoidance of manual tuning of closure parameters. For a detailed description of LESNIC, see Esau (2004a) and, for intercomparison, Fedorovich et al. (2004) and Beare et al. (2006). This code was used to perform a numerically and physically consistent set of experiments at 643 mesh, which comprised DATABASE64. It includes more than 160 runs with different combinations of governing parameters: surface heat flux, Brunt–V¨ais¨al¨a frequency in the free atmosphere, wind speed and direction, Coriolis parameter, and surface roughness. All runs started from laminar flow perturbed with small random fluctuations and was computed over 16 model hours. This implied considerably longer period of equilibration than 5 model hours used earlier by Zilitinkevich and Esau (2003). Since the TN PBL regime requires the longest time for equilibration, one additional case of the TN PBL was run for 24 model hours to assure that the steady state has indeed been achieved within 10% of accuracy. Data from the last hour were used to calculate the steady-state turbulent statistics. In all runs the PBL comprised between 0.5 and 0.67 of the total domain depth. In the figures presented below the PBL height is determined from the condition |τ |z=h = 0.05 u2∗ . Copyright 2007 Royal Meteorological Society
For our purposes, the basic advantage of the DATABASE64 is that it guarantees completely controlled conditions of each run. This allows applying the following two-step strategy. First, we summarize information about hE in the TN PBL and in the two comparatively simple transition regimes: TN–CN (between the truly and the conventionally neutral PBLs, with Bs = 0) and TN–NS (between the truly neutral and the nocturnal stable PBLs, with N = 0); and by these means determine the constants CR , CCN and CNS . Then we substitute these constants into Equation (3) and verify it against all relevant LES data (including complex cases with Bs < 0 and N > 0 excluded at the first step) to evaluate quantitatively whether Equation (3) can or cannot be recommended for practical calculations of the PBL height. In the above transition regimes, Equation (3) reduces to the following formulae convenient for verification:
u∗ f hE
2
=
−2 CR−2 + CCN µN −2 −2 µ CR + CNS
for TN–CN for TN–NS.
(5a) (5b)
LES data on the TN PBL height presented in Figure 1 confirm that the dimensionless coefficient in the Rossby– Montgomery formula does not show systematic dependence on the surface Rossby number, Ro = Ug (|f |z0u )−1 (the only dimensionless parameter relevant to this regime), and indeed can be considered as a universal constant. Our new estimate (CR = 0.7 ± 0.1 based on several 16 model-hour LES experiments) is larger then the previous one (CR = 0.5; Zilitinkevich and Esau, 2003) based on a shorter LES run (5 model hours) in a more limited LES domain. For comparison, direct numerical simulation (DNS) of the TN PBL over a smooth surface gave CR = 0.625 ± 0.025 (Coleman, 1999), and more recently, for larger Reynolds numbers, CR = 0.605 (Coleman et al., 2005). An LES run with four times higher resolution (at the mesh 2563 ) gave CR = 0.55 ± 0.05. In the circumstances we recommend a reasonable compromise value of CR = 0.6. Figures 2 and 3 show LES data on the dimensionless squared reciprocal of the equilibrium PBL height, u2∗ (f hE )−2 , versus µN = N |f |−1 (in the TN–CN regime) and µ = u∗ |f L|−1 (in the TN–NS regime). They support Equation (5a,b) and give the constants CCN = 1.36 ± 0.25 and CNS = 0.51 ± 0.06 (determined through the root-mean-square fit for Ug > 4 m s−1 in the intervals 20 < µN < 120 and 0 < µ < 40, respectively). Higher uncertainty in the determination of the PBL height in the regimes close to the TN PBL is not surprising. Indeed, when the free flow is neutrally (or weakly stably) stratified, large eddies generated in the PBL interior easily extend quite far in the vertical and live quite long, which causes pronounced variations in the estimated PBL height. Hence very long runs are needed to achieve stable estimates of CR . Taking the above values of empirical constants: CR = 0.6, CCN = 1.36 and CNS = 0.51, Equation (3) allows Q. J. R. Meteorol. Soc. 133: 265–271 (2007) DOI: 10.1002/qj
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S. ZILITINKEVICH ET AL. 1.5
CR = f hEu*−1
1
0.5
0
3
3.5
4
4.5
5
5.5 log (Ro)
6
6.5
7
7.5
8
Figure 1. Empirical constant CR in the Rossby–Montgomery formula for the height of the TN PBL versus the surface Rossby number Ro = Ug (|f |z0u )−1 , from our LES. Solid line shows the mean value of CR ; dotted lines show one standard deviation. 100 90 80 70
u2* (f hE)−2
60 50 40 30 20 15 m s−1 5 m s−1
10 0
0
50
100
150
µN = N f −1
Figure 2. LES validation of Equation (5a) for the equilibrium PBL height, hE , in the transition regime covering TN and CN PBLs. The sizes of circles indicate the geostrophic wind speeds in different LES runs. The theoretical line shows Equation (5a) with CR = 0.6 and CCN = 1.36 (the best RMS fit over the interval 20 < µN < 120, for Ug > 4 m s−1 ).
calculation of the PBL height, htheory , in all PBL regimes represented in our database (including numerous independent data for Bs < 0, N > 0 not used in the above analysis of the TN–CN and TN–NS transition regimes). Copyright 2007 Royal Meteorological Society
Figure 4 shows comparison of this htheory with the PBL height, hLES , directly obtained from LES (from the condition |τ |z=h = 0.05 u2∗ ), and confirms quite good performance of Equation (3). Q. J. R. Meteorol. Soc. 133: 265–271 (2007) DOI: 10.1002/qj
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EQUILIBRIUM HEIGHT OF NEUTRAL AND STABLE BOUNDARY LAYERS 200
180
160
140
u*2 (f hE)−2
120
100
80
60
40 10 m s−1 5 m s−1
20
0
0
5
10
15
20
25
30
35
40
45
50
µ = u* ( f L )−1
Figure 3. LES validation of Equation (5b) for the equilibrium PBL height, hE , in the transition regime covering TN and NS PBLs. The sizes of the circles indicate the geostrophic wind speeds. The theoretical line shows Equation (5b) with CR = 0.6 and CNS = 0.51 (the best RMS fit over the interval 0 < µ < 40, for Ug > 4 m s−1 ). 3.5
log10 htheory
3
2.5
2
1.5 1.5
2
15 m s−1
LS regime
10 m s−1
NS regime
5 m s−1 2 m s−1
CN regime
2.5 log10 hLES
TN regime 3
3.5
Figure 4. Correlation between the PBL heights: htheory calculated from Equation (3) (with CR = 0.6, CCN = 1.36 and CNS = 0.51) and hLES directly obtained from LES. The sizes of circles indicate the geostrophic wind speeds. Black circles represent complex long-lived stable (LS) PBL regimes not included in our estimates of CR , CCN and CNS , and not presented in Figures 1–3.
Copyright 2007 Royal Meteorological Society
Q. J. R. Meteorol. Soc. 133: 265–271 (2007) DOI: 10.1002/qj
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Concluding remarks
The equilibrium height, hE , of the stable and neutral PBLs depends on many contradicting factors which requires multi-limit theoretical analyses leading to complex PBL height formulations. Some parameters controlling the PBL height, in particular, the baroclinic shear, , and the large-scale vertical velocity at the PBL upper boundary, wh , as well as the PBL height as such, are quite difficult to measure in the atmosphere and the ocean. Moreover atmospheric and oceanic PBLs are almost always non-stationary and horizontally heterogeneous. As a result, data from currently available meteorological and oceanographic experiments happen to be not the best material for empirical validation of theoretical hE equations. In this situation, data on hE under completely controlled conditions obtained from LES or DNS are very useful. The latter can be treated as reasonable imitations of field experiments, provided that they realistically reproduce all other essential features of the PBL. In our recent work, we systematically used LES to validate theoretical hE formulations, and supported these analyses through independent comparison of our LES with experimental data. By this means we have derived and validated a multi-limit PBL height formulation, Equation (3), which reduces to known asymptotic limits in the truly neutral, conventionally neutral and nocturnal stable PBL regimes. For lack of a better empirically grounded formulation, Equation (3) can be recommended for practical use in the steady-state barotropic regimes. Accounting for baroclinicity, Equation (3) should be corrected by substituting u2T = u2∗ (1 + C0 /N ) for u2∗ in the second term on the right-hand side (C0 ≈ 0.67 is an LES-based empirical constant; see Zilitinkevich and Esau 2003). Accounting for large-scale vertical motions, we recommend the following corrected PBL height formulation: hE−cor = hE + wh tT , where tT = CE hE /u∗ is a PBL relaxation time-scale, and CE is a dimensionless empirical constant (CE ≈ 1, see Zilitinkevich and Baklanov, 2002). Our Figures 2 and 3 give convincing confirmation of the theoretically predicted linear dependences of u2∗ (f hE )−2 on µN = N/|f | or µ = u∗ |f L|−1 in the TN–CN or TN–NS transition regimes, respectively. We recommend these well-established dependencies as test beds for comparison and verification of turbulent closure models in operational use. It is obvious that realistic closures should be capable of reproducing the PBL height, in particular, in the above regimes. Good correspondence between results from a turbulent closure model and data from our Figures 2 and 3 would be a strong confirmation of the high quality of the closure in question. This method seems to be more efficient than traditional case-studies, because our figures give strongly compressed information (since each point corresponds to an independent case-study). Copyright 2007 Royal Meteorological Society
Acknowledgements This work has been supported by the EU Marie Curie Chair Project MEXC-CT-2003-509742, EU Project FUMAPEX EVK4-2001-00281, ARO Project W911NF05-1-0055, Norwegian project MACESIZ 155945/700, joint Norwegian–USA project ROLARC 151456/720, and the NORDPLUS Neighbour 2005–2006 Project FI-51. References Baklanov A. 2002. Parameterisation of SBL height in atmospheric pollution models. Pp. 415–424 in Air Pollution Modelling and its Application XV. Borrego C, Schayes G (eds.), Kluwer Academic/Plenum Publishers: New York. Beare RJ, MacVean MK, Holtslag AAM, Cuxart J, Esau I, Golaz J-C, Jimenez MA, Khairoutdinov M, Kosovic B, Lewellen D, Lund TS, Lundquist JK, McCabe A, Moene AF, Noh Y, Raasch S, Sullivan P. 2006. An intercomparison of large-eddy simulations of the stable boundary layer. Boundary-Layer Meteorol. 118: 247–272. Coleman GN. 1999. Similarity statistics from a direct numerical simulation of the neutrally stratified planetary boundary layer. J. Atmos. Sci. 56: 891–900. Coleman GN, Johnstone R, Ashworth M. 2005. ‘DNS of the turbulent Ekman layer at Re = 2000’. In proceedings of ERCOFTAC Workshop: Direct and Large-Eddy Simulation-6, 12–14 September, Poitiers, France. Ekman VW. 1905. On the influence of the Earth’s rotation on ocean currents. Arkiv for Matematik, Astronomi och Fysik 2: 1–52. Esau I. 2004a. Simulation of Ekman boundary layers by large-eddy model with dynamic mixed sub-filter closure. Environ. Fluid Mech. 4: 273–303. Esau I. 2004b. An improved parameterization of turbulent exchange coefficients accounting for the non-local effect of large eddies. Ann. Geophys. 22: 3353–3362. Fedorovich E, Conzemius R, Esau I, Chow FK, Lewellen D, Moeng C-H, Pino D, Sullivan P, Vila-Guerau de Arellano J. 2004. ‘Entrainment into sheared convective boundary layers as predicted by different large-eddy simulation codes’. In proceedings of 16th AMS Symposium on Boundary Layers and Turbulence. http://ams.confex.com/ams/pdfpapers/78656.pdf. Kitaigorodskii SA, Joffre SM. 1988. In search of a simple scaling for the height of the stratified atmospheric boundary layer. Tellus A 40: 419–433. Mason PJ, Thomson DJ. 1987. Large-eddy simulations of the neutralstatic-stability planetary boundary layer. Q. J. R. Meteorol. Soc. 113: 413–443. Mason PJ, Thomson DJ. 1992. Stochastic backscatter in large-eddy simulations of boundary layers. J. Fluid Mech. 242: 51–78. Monin AS, Obukhov AM. 1954. Basic laws of turbulent mixing in the ground layer of the atmosphere. Trans. Geophys. Inst. Akad. Nauk. USSR 151: 163–187. Piringer M, Joffre S. (Eds.) 2005. ‘The urban surface energy budget and mixing height in European cities: Data, models and challenges for urban meteorology and air quality’. Final Report of Working Group 2 of COST-715 Action. Demetra Ltd., Sofia, Bulgaria. Pollard RT, Rhines PB, Thompson R. 1973. The deepening of the wind-mixed layer. Geophys. Fluid Dyn. 3: 381–404. Rossby CG, Montgomery RB. 1935. The layer of frictional influence in wind and ocean currents. Papers in Phys. Oceanogr. Meteorol. (MIT and Woods Hole Oceanogr. Inst.) 3: 1–101. Seibert P, Beyrich F, Gryning S-E, Joffre S, Rasmussen A, Tercier P. 2000. Review and intercomparison of operational methods for the determination of the mixing height. Atmos. Env. 34: 1001–1027. Steeneveld GJ, van de Wiel BJH, Holtslag AAM. 2007. Comments on deriving the equilibrium height of the stable boundary layer. Q. J. R. Meteorol. Soc. 133: in press. Troen I Mahrt L. 1986. A simple model of the atmospheric boundary layer: Sensitivity to surface evaporation. Boundary-Layer Meteorol. 37: 129–148. Zilitinkevich SS. 1972. On the determination of the height of the Ekman boundary layer. Boundary-Layer Meteorol. 3: 141–145. Zilitinkevich S. 2002. Third-order transport due to internal waves and non-local turbulence in the stably stratified surface layer. Q. J. R. Meteorol. Soc. 128: 913–925. Q. J. R. Meteorol. Soc. 133: 265–271 (2007) DOI: 10.1002/qj
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