Mesoscale Ascent in Nocturnal Low-Level Jets

Alan Shapiro; Evgeni Fedorovich; Joshua G. Gebauer

doi:10.1175/JAS-D-17-0279.1

Jump to Content Jump to Main Navigation

Journal of the Atmospheric Sciences

Abstract
1. Introduction
2. Laterally varying free-atmosphere geostrophic wind
- a. Late afternoon profiles of wind and buoyancy
- b. Postsunset evolution for the inhomogeneous -forced problem
3. Laterally varying CBL buoyancy
- a. Late afternoon profiles of wind and buoyancy
- b. Postsunset evolution for the inhomogeneous -forced problem
4. Mesoscale ascent in LLJs during PECAN
5. Summary and final remarks
Bessel Functions of the First Kind
Approximating the Forcing Term over a Shallow Slope

ProCite

RefWorks

Reference Manager

BibTeX

Zotero

EndNote

View raw image

Schematic of wind and potential temperature profiles in the CBL (thin lines) and the corresponding profiles in a zero-order model of the CBL (bold lines). Variables in the zero-order model jump in value across a capping inversion at . Adapted from Fedorovich (1995).

View raw image

Time–height plot of (cm s⁻¹) from (2.37) for the -forced problem with , , H = 2 km, and A = 5 m s⁻¹.

View raw image

Schematic of buoyancy b and ascent in a warm tongue. The magnitude of b increases with shading intensity; red indicates positive b, blue indicates negative b. The baroclinic term in the horizontal vorticity equation [(2.21)] generates negative where , and positive where . The associated horizontal convergence and ascent are largest over the buoyancy maximum. In the Northern Hemisphere, the Coriolis force can be expected to gradually deflect the flow northward east of this maximum and southward west of this maximum.

View raw image

Time–height diagrams of (cm s⁻¹) from (3.14) for CBL-buoyancy-forced cases with wavelengths (left) and (right) , and buoyancy frequencies (top) , (middle) , and (bottom) . Other parameter values are given in Table 1.

View raw image

Vertical cross sections of (left) initial buoyancy (m s⁻²) and (right) initial potential temperature (K) for the CBL-buoyancy-forced case corresponding to the middle-left panel of Fig. 4.

View raw image

Evolution of w (cm s⁻¹) for the CBL-buoyancy-forced case corresponding to the middle-left panel of Fig. 4. The initial buoyancy field is shown in Fig. 5. (top)–(bottom) Time increases at 1-h intervals from t = 0.5 to 5.5 h.

View raw image

Domainwide maximum (left) vertical velocity and (right) vertical parcel displacement as functions of N for (solid line) and (dashed line). Other parameter values are given in Table 1.

View raw image

As in Fig. 7, but as a function of latitude.

View raw image

As in Fig. 7, but as a function of H.

View raw image

Atmospheric conditions on 2 Jun 2015: (a) 0000 UTC HRRR 2-m temperature analysis; (b) 0000 UTC NOAA/NWS/Storm Prediction Center (SPC) 700-hPa height, temperature, and wind analysis; (c) S-Pol radial winds on the 0.5° scan surface at (left) 0009, (center) 0406, and (right) 0800 UTC; and (d) radar reflectivity over KS and southern NE from the NWS National Mosaic at (left) 0908 and (right) 1208 UTC. Range rings (thin circles) in (c) are plotted at 50-km intervals. The left panel of (d) shows locations of S-Pol (dot), Ellis (square), and Hays (diamond).

View raw image

Evolution of selected profiles on 2 Jun 2015: (a) WV-DIAL relative backscatter, (b) 449-MHz profiler signal-to-noise ratio (dB), and (c) MPL relative backscatter (C km² μs⁻¹) at Ellis.

View raw image

Atmospheric conditions on 14 Jun 2015: (a) 0000 UTC HRRR 2-m temperature analysis; (b) 0000 UTC NOAA/NWS/SPC 700-hPa height, temperature, and wind analysis; (c) S-Pol radial winds on the 0.5° scan surface at (left) 0009, (center) 0403, and (right) 0807 UTC; and (d) radar reflectivity over KS from the NWS National Mosaic at (left) 0508 and (right) 1008 UTC. Range rings (thin circles) in (c) are plotted at 50-km intervals. The left panel of (d) shows locations of S-Pol (dot), Ellis (square), and Hays (diamond).

View raw image

Evolution of selected profiles on 14 Jun 2015: (a) WV-DIAL relative backscatter and (b) 449-MHz profiler signal-to-noise ratio (dB) at Ellis and (c) CL31 ceilometer backscatter [(10⁵ srad km)⁻¹] at the S-Pol location.

View raw image

Atmospheric conditions on 5 Jul 2015: (a) 0000 UTC HRRR 2-m temperature analysis; (b) 0000 UTC NOAA/NWS/SPC 700-hPa height, temperature, and wind analysis; (c) S-Pol radial winds on the 0.5° scan surface at (left) 0005, (center) 0406, and (right) 0807 UTC; and (d) radar reflectivity over KS from the NWS National Mosaic at (left) 0358 and (right) 0758 UTC. Range rings (thin circles) in (c) are plotted at 50-km intervals. The left panel of (d) shows locations of S-Pol (dot), Ellis (square), and Hays (diamond).

View raw image

Evolution of selected profiles on 5 Jul 2015: (a) WV-DIAL relative backscatter, (b) WV-DIAL absolute humidity (g m⁻³), and (c) MPL relative backscatter (C km² μs⁻¹) at Ellis.

View raw image

700-hPa GFS analysis of ω at 0600 UTC 5 Jul 2015. Figure courtesy of Dylan Reif.

	All Time	Past Year	Past 30 Days
Abstract Views	0	0	0
Full Text Views	342	237	25
PDF Downloads	262	166	20

Chart III. Departure (°F.) of the Mean Temperature from the Normal, July, 1922

DISCUSSION

STORMS AND WEATHER WARNINGS

METEOROLOGICAL CENTERS OF ACTION IN THE NORTH PACIFIC OCEAN.

Author: EDWARD A. BEALS

GENERAL CONDITIONS

Author: A. J. HENRY

Editorial Type:: Article

Mesoscale Ascent in Nocturnal Low-Level Jets

Alan Shapiro¹, Evgeni Fedorovich², and Joshua G. Gebauer³

View More View Less

¹ School of Meteorology, and Center for Analysis and Prediction of Storms, University of Oklahoma, Norman, Oklahoma
| ² School of Meteorology, University of Oklahoma, Norman, Oklahoma
| ³ School of Meteorology, and Cooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, Norman, Oklahoma

Print Publication:: 01 May 2018

Collections:: Plains Elevated Convection At Night (PECAN)

DOI:: https://doi.org/10.1175/JAS-D-17-0279.1

Page(s):: 1403–1427

Received:: 18 Sep 2017

Final Form:: 12 Feb 2018

Published Online:: May 2018

Download PDF

Full access

Abstract

A theory for gentle but persistent mesoscale ascent in the lower troposphere is developed in which the vertical motion arises as an inertia–gravity wave response to the sudden decrease of turbulent mixing in a horizontally heterogeneous convective boundary layer (CBL). The zone of ascent is centered on the local maximum of a laterally varying buoyancy field (warm tongue in the CBL). The shutdown also triggers a Blackadar-type inertial oscillation and associated low-level jet (LLJ). These nocturnal motions are studied analytically using the linearized two-dimensional Boussinesq equations of motion, thermal energy, and mass conservation for an inviscid stably stratified fluid, with the initial state described by a zero-order jump model of a CBL. The vertical velocity revealed by the analytical solution increases with the amplitude of the buoyancy variation, CBL depth, and wavenumber of the buoyancy variation (larger vertical velocity for smaller-scale variations). Stable stratification in the free atmosphere has a lid effect, with a larger buoyancy frequency associated with a smaller vertical velocity. For the parameter values typical of the southern Great Plains warm season, the peak vertical velocity is ~3–10 cm s⁻¹, with parcels rising ~0.3–1 km over the ~6–8-h duration of the ascent phase. Data from the 2015 Plains Elevated Convection at Night (PECAN) field project were used as a qualitative check on the hypothesis that the same mechanism that triggers nocturnal LLJs from CBLs can induce gentle but persistent ascent in the presence of a warm tongue.

Corresponding author: Alan Shapiro, ashapiro@ou.edu

This article is included in the Plains Elevated Convection At Night (PECAN) Special Collection.

Abstract

Corresponding author: Alan Shapiro, ashapiro@ou.edu

This article is included in the Plains Elevated Convection At Night (PECAN) Special Collection.

Keywords: Dynamics; Mesoscale processes; Boundary layer; Jets

1. Introduction

It has long been recognized that warm season precipitation over the central United States exhibits a nocturnal maximum (Kincer 1916; Bleeker and Andre 1951; Means 1952; Pitchford and London 1962; Wallace 1975; Easterling and Robinson 1985; Riley et al. 1987; Dai et al. 1999; Carbone et al. 2002; Carbone and Tuttle 2008). This rainfall is beneficial for agriculture, but it is also associated with lightning, flooding, and other weather hazards (Crysler et al. 1982; Orville and Henderson 1986; Maddox et al. 1979, 1986; Fritsch et al. 1986; Jirak and Cotton 2007). Operational numerical weather prediction and global circulation models have little skill in forecasting nocturnal precipitation in this region (Davis et al. 2003; Clark et al. 2007; Lee et al. 2008; Surcel et al. 2010; Song et al. 2013).

Nocturnal convection over the central United States often develops within an eastward-propagating envelope of successively dissipating and regenerating mesoscale convective systems (MCSs) originating from afternoon thunderstorms over the Rocky Mountains (Maddox 1983; Augustine and Caracena 1994; Dai et al. 1999; Carbone et al. 2002; Davis et al. 2003; Carbone and Tuttle 2008). However, it can also develop without any apparent connection to ongoing or preexisting convection (Wilson and Roberts 2006; Reif and Bluestein 2017; Geerts et al. 2017). During the International H₂O Project (IHOP_2002) over the southern Great Plains of the United States, more than half of the 26 storm complexes that entered the study area dissipated within the domain, while 112 cases of in situ convection initiation (CI) occurred (Wilson and Roberts 2006). Approximately half of these CI cases were triggered without the presence of a nearby surface convergence boundary and occurred mostly at night. Many of these elevated nocturnal CI cases were associated with regions of synoptic or mesoscale convergence in the 900–600-hPa layer apparent in Rapid Update Cycle (RUC) wind analyses. In a 20-yr climatology of warm season nocturnal CI over the central and southern Great Plains, Reif and Bluestein (2017) found that 24% of the nocturnal CI episodes occurred without a nearby surface boundary. Nearly one-half of these no-boundary (NB) CI episodes were of a linear storm type, the majority of which had a preferred north–south orientation, the same preference exhibited by nocturnal low-level jets (LLJs) over the Great Plains (e.g., Hoecker 1963; Bonner 1968; Bonner et al. 1968; Mitchell et al. 1995; Whiteman et al. 1997; Song et al. 2005). Reif and Bluestein (2017) found a common association of LLJs with all of the nocturnal CI modes, but noted that the association was closest with the NB mode.

Regions of low- or midlevel horizontal convergence in the nocturnal environment can promote or initiate convection by deepening the moist layer and lifting air parcels toward their level of free convection (Weckwerth and Parsons 2006; Wilson and Roberts 2006). Most recently, Trier et al. (2017) suggested that weak but persistent mesoscale ascent might lead to the formation of moist absolutely unstable layers (MAULs), from which convection can develop. Elucidating the mechanisms of nocturnal CI can be challenging because of the subtleties and complexities of the many candidate phenomena and the fact that many of these phenomena occur simultaneously. Among the plausible mechanisms for CI are convergence and ascent associated with deep-tropospheric gravity waves (e.g., Uccelini 1975; Koch et al. 1988; Fovell et al. 2006; Marsham and Parker 2006); cold fronts, density currents, and drylines (e.g., Charba 1974; Wilson and Schreiber 1986; Mahoney 1988; Weckwerth and Wakimoto 1992; Hane et al. 1993; Ziegler and Rasmussen 1998; Weiss and Bluestein 2002; Geerts et al. 2006; Weckwerth et al. 2008); inland or “vegetation” breezes (e.g., Sun and Ogura 1979; Mahfouf et al. 1987; Segal and Arritt 1992; Mahrt et al. 1994; Lynn et al. 1998; McPherson 2007; Drobinski and Dubos 2009); bores, solitons, and other shallow trapped gravity waves (e.g., Carbone et al. 1990; Karyampudi et al. 1995; Koch et al. 1988; Koch and Clark 1999; Coleman and Knupp 2011; Marsham et al. 2011; Haghi et al. 2017); and nocturnal LLJs (e.g., Means 1952; Blackadar 1959; Pitchford and London 1962; Bonner 1966; Bonner et al. 1968; Paegle and Rasch 1973; Paegle and McLawhorn 1973; Paegle 1978; Maddox 1983; Maddox and Grice 1983; Wallace 1975; Astling et al. 1985; Trier and Parsons 1993; Augustine and Caracena 1994; Higgins et al. 1997; Arritt et al. 1997; Walters and Winkler 2001; Tuttle and Davis 2006; Trier et al. 2006, 2014, 2017; Carbone and Tuttle 2008; French and Parker 2010; Pu and Dickinson 2014; Reif and Bluestein 2017). Some of the latter LLJ studies also suggest that there are multiple modes of LLJ-associated convection: nocturnal convection over the Great Plains is most often initiated or maintained by convergence at the jet terminus (northern terminus in the typically southerly jet) or at the intersection of a jet with a front or other surface boundary, but is also often observed along a lateral flank of a jet.

In this study we explore a boundary layer mechanism for inducing gentle but persistent mesoscale^¹ ascent in warm season LLJs. We do not address CI per se but consider a mechanism that produces net vertical parcel displacements of magnitudes that may facilitate CI. Our work is motivated by the close associations between LLJs and nocturnal convection over the Great Plains described in many of the above-noted LLJ papers and also documented for other regions worldwide (e.g., Velasco and Fritsch 1987; Stensrud 1996; Liebmann et al. 2004; Salio et al. 2007; Monaghan et al. 2010; Wang et al. 2013; Chen and Tomassini 2015). The roles of ascent in a decelerating flow north of an LLJ wind maximum (in a southerly LLJ), or along or north of the intersection of an LLJ with a cold front, are more or less clear, but the mechanisms that force ascent on a lateral flank of an LLJ are still not well understood. In this regard, we believe that the recent Pu and Dickinson (2014) explanation for such a mechanism is not wholly satisfactory. In a study of vertical motions in Great Plains LLJs using a North American Regional Reanalysis (NARR) June–July climatology, Pu and Dickinson (2014) suggest that after midnight, as the jet weakens, the vertical vorticity should increase (decrease) to the east (west) of the jet core,^² and that vertical motions “are developed to balance” those tendencies. However, a balance concept may not be appropriate for Great Plains LLJs (no justification was given), and the authors’ vertical vorticity budget analysis merely indicates that diurnal changes in the vertical vorticity over the scale of their analysis domain are largely due to the stretching of Earth vorticity, regardless of the convergence mechanism.

We use an analytical model to explore how weak but persistent convergent motions can arise from the release of the frictional constraint in a horizontally inhomogeneous convective boundary layer (CBL) at sunset. To illustrate the concept in its most basic form and gain some insights into the nature and scale of the kinds of inhomogeneities that may be relevant, we work with an idealized zero-order jump model of the CBL for the initial state and a simple inviscid model for the nocturnal motion. We consider, in turn, lateral variations in the free-atmosphere geostrophic wind and CBL buoyancy. The nocturnal state following the shutdown of mixing is modeled as a two-dimensional (2D) inviscid flow of a stably stratified fluid. In this scenario, flow convergence cannot occur at the terminus of a jet (there is no terminus) but is parallel to the jet axis. The shutdown of mixing that triggers the convergence in our theory is the same mechanism that triggers an inertial oscillation (IO) that manifests as an LLJ in Blackadar’s (1957) pioneering theory.^³ Both modes of motion occur in our model, but because the initial wind field is independent of height in the CBL (well-mixed assumption in the zero-order model), the nocturnal IO–LLJ is slab-like.

In section 2 we present the governing equations. Solutions are obtained for initial states characterized by lateral variations in free-atmosphere geostrophic wind (section 2) and CBL buoyancy (section 3). As the latter variations are found to be much more effective in generating vertical motions, we focus on the relevance of buoyancy forcing in three CI cases from the Plains Elevated Convection at Night (PECAN; Geerts et al. 2017) field project (section 4). A summary and concluding remarks follow in section 5.

2. Laterally varying free-atmosphere geostrophic wind

a. Late afternoon profiles of wind and buoyancy

Our initial () state is based on a zero-order jump model of a deep late afternoon CBL with overlying free atmosphere (Fedorovich 1995), schematized in Fig. 1. In the zero-order model, the bulk of the CBL is idealized as a layer of strong turbulent mixing (mixed layer) throughout which momentum and potential temperature are independent of height z. The top of the CBL coincides with a capping inversion of vanishing thickness at across which the wind and potential temperature jump in value. For vertical integrals up to in which a distinction between the top or bottom of the inversion is important, we denote the upper limit of integration by if the inversion is included and by if the inversion is excluded.

We consider the free atmosphere above the CBL to be barotropic (free-atmosphere geostrophic wind is independent of z), hydrostatic, and in a geostrophic balance. The free-atmosphere geostrophic wind is aligned with the y axis (pointing north)^⁴ and varies in the east–west x direction as

where A is the amplitude of the variation, k is the wavenumber, and

is the horizontal mean value. Without loss of generality, we take k and A to be positive.

The free-atmosphere potential temperature

increases linearly with height, so the free-atmosphere buoyancy (Brunt–Väisälä) frequency N, defined through

is constant. Here g is the acceleration due to gravity and

(= 300 K) is a constant reference value of potential temperature.

We assume the y-component wind

varies as a piecewise constant function of z, with a discontinuity at the capping inversion. Within the CBL,

is a constant fraction

(<1) of the free-atmosphere geostrophic wind, while above the CBL,

is equal to the free-atmosphere geostrophic wind:

For later use, we note that (2.3) implies

where

is the Dirac delta function. The x-velocity component

is set to zero everywhere, that is, we neglect the frictional-stress-mediated (Ekman) cross-isobar flow. In CBLs this component is often much smaller than the component in the direction of the geostrophic wind (Arya 1977; van Ulden and Holtslag 1985). The vertical velocity

is set to zero everywhere.

We consider the potential temperature

to be independent of x and z within the CBL, to jump in value across the capping inversion, and to follow the free-atmosphere profile

above the CBL:

Here

is the potential temperature jump (inversion strength). Integration of (2.2) yields

The

profile defined by (2.6) coincides with the actual potential temperature profile in the free atmosphere (z > H). However, we use this same (2.6)—for all z—to define the potential temperature for the reference atmosphere. This reference profile appears in (2.5) and in the definition of buoyancy,

Applying (2.5) and (2.6) in (2.7) yields the late afternoon buoyancy profile as

The kinematic pressure perturbation

[

, where

is pressure,

is the reference pressure profile at a fixed x location

(so

), and

is a constant value of density] satisfies the hydrostatic equation in the initial state (

) and in the postsunset motion (

). Since

is independent of x,

is independent of x. Equivalently,

is independent of z. We therefore determine

at any height from its distribution in the free atmosphere, which we consider to be in the geostrophic balance,

, where the Coriolis parameter f is constant. In view of this balance and (2.1), the integrated hydrostatic equation yields

b. Postsunset evolution for the inhomogeneous -forced problem

We investigate the atmospheric response to the release of the frictional constraint at sunset using the 2D linearized Boussinesq equations of motion, thermal energy, and mass conservation for an inviscid stably stratified fluid on an f plane:

In view of (2.3), (2.9), and (2.10), the time-dependent part of u has

and horizontally uniform components. Equation (2.11) then shows that the time-dependent part of υ also has

and horizontally uniform components, while (2.14) shows that w varies in x as

. Equation (2.13) then shows that the time-dependent part of b varies in x as

, in which case (2.12) [or (2.10)] shows that the time-dependent part of

also varies as

. We thus write

where a subscript m denotes an association with a horizontally uniform but time dependent motion, and a subscript 0 denotes the initial (late afternoon) state of a variable, as defined in section 2a. Note that

represents the w profile at the x location of peak strength, that is, where

. Since the initial states of u, υ, w, b, and

appear as the 0 subscript variables in (2.15a)–(2.15e), the m and k subscript variables are initially zero:

When (2.15a)–(2.15e), (2.3), (2.8), and (2.9) are applied in (2.10)–(2.14), which are then averaged in x over one wavelength λ (

), the only surviving terms involve the horizontally uniform time-dependent variables, which satisfy

and the late afternoon buoyancy and perturbation pressure terms, which satisfy the hydrostatic equation. The solution of (2.17) and (2.18) subject to (2.16a) is

The sum of this (horizontally homogeneous) solution and the horizontally homogeneous part of the initial state is a Blackadar-type IO. We will refer to this motion as the homogeneous jet. The peak wind speed in the homogeneous jet occurs at time

, when the ageostrophic wind aligns with the free-atmosphere geostrophic wind (southerly in this case). This maximum homogeneous jet speed is the sum of the peak

value of

and the initial (

) contribution

. In the extreme (unrealistic) case where friction has brought the late afternoon southerly wind component in the mixed layer to zero (

), the peak wind speed would be twice the free-atmosphere geostrophic wind

. For more realistic

, the peak speed would be much less than this bound. For example, for

the peak speed is

, which is still supergeostrophic, but only exceeds the geostrophic value by 30%.

We now turn to the laterally variable part of the flow and the associated vertical motion. Taking

yields an equation for the production of y-component vorticity (

Taking

of (2.21) and using (2.11) and (2.13) to eliminate the tendencies of υ and b yields

Taking

of (2.22) followed by the use of (2.14) to eliminate u yields a single equation for w,

Equation (2.23) is a well-known wave equation for motions in a hydrostatic inertia–gravity wave regime (e.g., Gill 1982, p. 260; Lin 2007, p. 47). It admits free oscillations of the form

, where W is the amplitude, m is the vertical wavenumber, and the frequency

satisfies the dispersion relation

. Since the vertical component of the group velocity is

, upward energy propagation (

) requires that

and m are of opposite sign, in which case the phase propagation has a downward component. The time scales in this regime are about an order of magnitude smaller than the time scales in quasigeostrophic motions.

Applying (2.15c) in (2.23) yields

We solve (2.24) using the method of Laplace transforms (e.g., Doetsch 1961). Multiplying (2.24) by

, where s is a complex frequency, and integrating the resulting equation over time yields

where

is the Laplace transform of

and

In view of (2.16b), the

term in (2.26) is zero. Taking

of (2.21) at

and then using (2.14) to eliminate u in favor of w yields the tendency term in (2.26) as

Evaluating this tendency term using (2.4) and (2.8), and applying it in (2.26), we obtain

We solve (2.25) subject to the impermeability condition,

and the remote condition,

The method of variation of parameters yields the general solution of (2.25):

Applying (2.28) in (2.31) yields

Applying (2.29) and (2.30) in (2.32) yields

We then obtain

(for all z) as

where we have used

for

, and defined

Combining (5) on p. 227 with (40) on p. 245 of Erdélyi et al. (1954), yields the inverse Laplace transform

for c > 0 in the form:

where

are Bessel functions of the first kind (appendix A). Using (2.36), we obtain the inverse transform of (2.34) as

Inspection of (2.37) shows that is proportional to , which is the amplitude of the step change in the laterally varying part of . The dependences of on k, N, and f are not as clear since those parameters appear in the arguments of the Bessel functions. However, numerical evaluation of (2.37) (appendix A) reveals that increases with increasing k, increasing , and decreasing .

Numerical evaluation of (2.37) for a wide range of parameters shows that the vertical motion is quite weak, with peak values generally less than 1 cm s⁻¹. An example is shown in Fig. 2 for the case where s⁻¹, s⁻¹, m, , m s⁻¹, m s⁻¹, and km ( m⁻¹); here barely reaches 0.5 cm s⁻¹. Horizontal variations in the free-atmosphere geostrophic wind are ineffective in generating ascent.

3. Laterally varying CBL buoyancy

a. Late afternoon profiles of wind and buoyancy

Our initial state is again described by a zero-order jump model of a CBL with a barotropic free atmosphere above it. However, now the potential temperature is considered to vary laterally within the CBL while the free-atmosphere geostrophic wind is horizontally homogeneous. Again we set

and

to zero. The southerly wind

is again a piecewise constant function of height with jump across the capping inversion, but now without an x dependence:

The laterally varying potential temperature

is well mixed vertically within the CBL, jumps in value

across the capping inversion, and follows the free-atmosphere profile of

above the CBL. The inversion strength and

profile are specified as

where

is the amplitude of the potential temperature variation (a negative value of

corresponds to a warm tongue centered on

), and

is the horizontal mean of

. We can specify

in terms of the inversion strength maximum

and minimum

. For the top of the CBL to be statically stable (

and

must be prescribed such that

. The

dependence chosen for

allows the phase relations in (2.15a)–(2.15e) to apply (with minor modifications), so we can reuse much of the mathematical development from section 2.

Applying (3.3) and (2.6) in (2.7) yields the buoyancy as

A schematic that depicts this buoyancy field, the baroclinic generation of horizontal vorticity, and the ascent realized after the shutdown of turbulence (i.e., when the horizontal vorticity is no longer mixed out) is shown in Fig. 3.

Integrating the hydrostatic equation with respect to z using

from (3.4), and determining the resulting function of integration using the free-atmosphere geostrophic balance with constant geostrophic wind

, we obtain

b. Postsunset evolution for the inhomogeneous -forced problem

The phase relations in (2.15a)–(2.15e) from the previous

-forced problem apply to this

-forced problem, though with (2.15a) and (2.15d) considered in the slightly modified forms,

When (2.10)–(2.14) are averaged over one wavelength, using (3.6a), (3.6b), (2.15b), (2.15c), and (2.15e) with the initial profiles in (3.1), (3.4), and (3.5), we again find that

and

satisfy (2.17) and (2.18) and have the same solution as in (2.19) and (2.20). Moreover, since

is the same as the homogeneous part of

in the

-forced problem, the homogeneous jets in the two problems are identical.

We now turn to the solution for

. An expression for

follows from (2.26), (2.27), (3.1), and (3.4) as

Applying (3.7) in (2.31) yields

Application of (2.29) and (2.30) in (3.8) produces

With these values, (3.8) becomes

which may be converted into a single form appropriate for all z:

Here

is the unit sign function (of

), and

and

are defined in (2.35).

Combining (5) on p. 227 with (36) on p. 245 of Erdélyi et al. (1954), we obtain the inverse transform

for

Using (3.12), we evaluate the inverse transform of (3.11) as

To bypass computational difficulties arising from the singularities of

and

in (3.13), we rewrite all Bessel functions having

in their arguments using (A.5), obtaining:

where

and we have used the fact that

for all z.

Since acts in (3.14) only through [via (3.15)], is directly proportional to . Less clear is how varies with H, k, and N, since those parameters appear in the arguments of the Bessel functions as well as in . However, numerical evaluation of (3.14) will show that the sensitivity of to these parameters is dominated by their appearance in , that is, increases as H and k increase and as N decreases.

The Bessel functions in (3.14) exhibit damped oscillations [e.g., see Fig. 9.1 of Abramowitz and Stegun (1964, hereafter AS)] whose properties are evident in (A.2), written here to leading order as

The zero-phase lines of the

functions with arguments

, and

are the zeroes of the cosine in (3.16). In view of (2.35), we see that these phase lines are at heights

, which descend with speeds

. Descending phase fronts are consistent with our discussion of (2.23), which also applies to these

-forced flows. For the

considered here, (3.16) also shows that larger changes in

are associated with higher-frequency oscillations, that is, oscillation periods decrease with larger N, larger k (smaller scale), and larger z. These inferences, however, only pertain to the first set of terms in (3.14). To more fully explore the characteristics of the motion, we evaluated (3.14) numerically using the procedure in appendix A.

Time–height plots of for the default parameters given in Table 1 and for a larger and smaller value of N ( respectively) are shown in Fig. 4. Not surprisingly, stratification provides a lid effect, with a larger N yielding a smaller . For both wavelengths considered, a ~50% reduction in N (bottom versus top panels) increases the peak by ~50%. For fixed N, the peak is larger for the smaller scale () forcing. For the typical free-troposphere value of (middle panels), a decrease in from 1000 to 600 km nearly doubles from about 5 to 10 cm s⁻¹. Consistent with our previous discussion, the pattern descends with time, and the duration of low-level ascent is longer for smaller N and larger λ. These results suggest competing effects of scale on vertical parcel displacements: a smaller λ increases net displacements by increasing , but decreases net displacements by shortening the ascent phase. As we will see, the former effect is more important than the latter, at least for the parameter values considered.

Table 1.

Default parameters for the CBL-buoyancy-forced experiments. The analytical solution [(3.14)] was evaluated numerically using the procedure outlined in appendix A with a vertical grid spacing of 50 m and a time interval of 300 s.

The case corresponding to the middle-left panel in Fig. 4 is examined in more detail. The initial buoyancy and potential temperature fields are shown in Fig. 5. The field is shown in Fig. 6. For the first ~1.5 h, the peak w is found slightly below the top of the preexisting CBL. The peak thereafter descends, though not as quickly as the phase line. Figure 4 shows that the largest vertical velocity observed at any time occurs at ~3 h, with low-level ascent lasting ~7 h. By integrating w in a Lagrangian sense, we found that , the largest net vertical parcel displacement of any air parcel originating within the CBL (parcels initially separated vertically by 100 m), was ~750 m. This displacement was attained near the end of the ascent phase for a parcel originating near the half-depth of the CBL.

Experiments were conducted in which one parameter at a time was varied from its default value. Figure 7 shows a strong increase in and with decreasing N. It should be borne in mind, however, that N in the free troposphere is typically very close to 0.01 s⁻¹ (Tsuda et al. 1991; Revathy et al. 1996; Yuan et al. 2010). Figure 8 shows an increase in and with decreasing latitude (i.e., decreasing f). Figure 9 shows a strong increase in and with increasing CBL depth. This result may be of special interest since warm season CBL depths over the southern Great Plains may exceed 2 km (Schmid and Niyogi 2012; Sawyer and Li 2013; Molod et al. 2015). Figures 7–9 show that and are larger for the smaller-scale forcing ().

Last, we consider the υ wind component. Recall that the laterally variable part of υ is

, where x = 0 corresponds to the center of ascent and the center of the warm tongue. In view of (2.4), (2.15b), (2.15c), and (2.11),

can be written as

This result also follows from the time-integrated vertical vorticity equation with forcing due to the stretching of Earth vorticity. Since the peak (positive)

occurs at a height between 1 and 2 km above the ground during much of the low-level ascent phase (Figs. 4, 6), and

is positive beneath that height, (3.17) indicates that the low-level

is positive and increases in magnitude throughout the ascent phase. Thus,

is positive (southerly) east of the center of ascent and negative (northerly) west of the center of ascent. These results are consistent with one’s intuition concerning the stretching of Earth vorticity (effect on lateral shear of υ), or the deflection of the u-wind component by the Coriolis force (effect on υ; see Fig. 3). Thus, for the typical case where the homogeneous jet is southerly, the peak ascent would be on the western flank of the full inhomogeneous jet. However, one can also envision a jet composed of a homogeneous jet and winds arising from both inhomogeneous

and

forcings, with a possible phase shift between the forcings. In such a case, the

forcing could affect the location of the jet maximum, but have little bearing on the location of the peak ascent (per results from section 2). If the phase shifts between the forcings were arbitrary, there should be no preference for the peak ascent to occur on either flank of the jet, but if the forcings were related, a preference could emerge.

4. Mesoscale ascent in LLJs during PECAN

Data from the PECAN project (Geerts et al. 2017) were used as a qualitative check on our hypothesis that the same mechanism that generates nocturnal jets can produce gentle but persistent nocturnal ascent in the presence of a mesoscale warm tongue. The PECAN campaign took place from 1 June to 15 July 2015, over central and western Kansas (KS) and the surrounding states, with an operations center in Hays, KS. Among the observational platforms were research aircraft, the NCAR Earth Observing Laboratory (EOL) S-Pol Ka radar (S-Pol), mobile Doppler radars, Doppler lidars, Raman lidars, ceilometers, micropulse lidars (MPLs), water vapor differential absorption lidars (WV-DIALs), sodars, wind profilers, and radiosondes. Data were gathered in 31 Intensive Observing Periods (IOPs) and 12 Unofficial Field Operations (UFOs),^⁵ each with a focus on MCSs, bores, CI, or LLJs. Only seven IOPs were LLJ or combined CI–LLJ missions, but LLJs were common throughout the campaign.

Several factors complicate comparisons of our theory with PECAN observations. Our model assumes a sudden and complete shutdown of turbulent mixing at sunset instead of a decrease of mixing to small but nonzero levels over the course of an evening transition that begins well before sunset (e.g., Acevedo and Fitzjarrald 2001; Wingo and Knupp 2015). The model does not take into account the Rocky Mountains ~250–300 km west of KS. The model does not make provision for terrain slope and thus cannot account for oscillations arising from the diurnal heating cycle of the sloping Great Plains (so-called Holton mechanism; Holton 1967; see also Fedorovich et al. 2017). Working with a simple analytical model, Du and Rotunno (2014) found that the amplitude and phase of their modeled Great Plains LLJs were in better agreement with data from the North American Regional Reanalysis when both the Holton and Blackadar (1957) mechanisms were accounted for. Shapiro et al. (2016) found that the relative strengths and timings of the LLJ wind maxima in their solutions accounting for Blackadar and Holton mechanisms were in qualitative agreement with the results in Du and Rotunno (2014). However, the former study also showed that while the Holton mechanism could act synergistically with the Blackadar mechanism to produce stronger jets, it was generally not as important as the Blackadar mechanism. An additional slope-related complication is that during the late afternoon, buoyancy increases westward on horizontal or isobaric surfaces that pass through the mixed layer into the surface layer as they obliquely approach the slope. The buoyancy increases with proximity to the heated ground, regardless of whether a warm tongue is present. Fortunately, as shown in appendix B, the forcing term in our theory can, in mixed layers over gentle slopes, be approximated in terms of variations of surface temperature. Use of a warm tongue in terms of temperature along the slope instead of mixed-layer buoyancy along a horizontal or isobaric surface allows us to bypass the above-noted difficulty. However, real surface temperatures inevitably have north–south variations and vary along the slope with more complexity than is represented by one harmonic. Rather than contrive a procedure to fit idealized 2D single-harmonic warm tongues to complex three-dimensional data, we identified warm tongues subjectively (i.e., visually).

A further difficulty is that the anticipated ascent rates () are beyond the accuracy ranges of the lidars and profilers deployed in PECAN and cannot be verified using w from those instruments. Instead, we estimated w indirectly (and admittedly crudely) by tracking the motion of layered structures in time–height plots^⁶ of data from the CL31 ceilometer (belonging to the University of Maryland, Baltimore County) at the S-Pol site (~40 km southwest of Hays), and data from the NCAR EOL 449-MHz profiler, Millersville University MPL, and NCAR EOL WV-DIAL at Ellis, KS (~25 km northwest of Hays), whenever such structures were evident (often they were not).

We identified four cases in which Blackadar-like LLJs, persistent weak rising motion, and pristine CI^⁷ developed at night after an afternoon in which conditions roughly conformed to the restrictions of our theory: 1 June (first day of PECAN, no IOP scheduled), 2 June (IOP1, a dry run), 14 June (UFO2, a CI mission), and 5 July (IOP19, a bore/MCS mission). The 1 June case was fairly similar to the 2 June case, and will not be described here. The CI episodes on 1 June, 2 June, and 5 July are discussed in Gebauer (2017). The 5 July case is also discussed in Reif and Bluestein (2017) and in Trier et al. (2017).

Overviews of conditions on 2 June 2015 are presented in Fig. 10 (see Figs. 12 and 14 for conditions on 14 June and 5 July 2015, respectively).^⁸ Surface and 700-hPa analyses at 0000 UTC showed quiescent synoptic conditions over KS, with any fronts located well north of the region. The 0000 UTC 2-m surface temperature analyses [from the NCEP High Resolution Rapid Refresh model (HRRR)] showed temperatures increasing westward across much of KS, with roughly north–south-oriented warm tongues centered over the western or west-central part of the state.

The 0000 UTC low-level base reflectivity fields from the NWS National Mosaic (not shown) indicated that KS was free of convective activity and mesoscale surface boundaries, while CBLs were evident in ~0000 UTC PECAN soundings^⁹ from Brewster, Ellis, and Greensburg (all in KS), and/or 0000 UTC NWS soundings from Dodge City, KS, and North Platte, Nebraska (NE), and/or 449-MHz profiler data from Ellis. Although the 700-hPa winds over central and western KS were weak on these afternoons (with a small northerly component evident on 2 June), the low-level thermal winds inferred from the 2-m temperature analyses were northerly, and the estimated low-level geostrophic winds had large southerly components. Against this backdrop of warm, dry afternoon conditions with southerly low-level geostrophic winds, it was not surprising to see the nocturnal development of southerly LLJs in Doppler velocity imagery from regional WSR-88Ds and the S-Pol radar. Figure 10c shows the evolution of the S-Pol radial winds for the 2 June case (see Figs. 12c and 14c for the 14 June and 15 July cases).^¹⁰ On each night, the wind speeds at ~0800 UTC at the level of the jet maximum (evident ~50 km from the S-Pol on the 0.5° tilt, which is ~400 m above the ground) were about twice their values at ~0000 UTC. Over the same period the wind vectors rotated clockwise from southeasterly to southwesterly.

From Global Telecommunications System (GTS) surface station plots at ~0000 UTC 2 June, 14 June, and 5 July, we estimate temperature changes of ~22°, ~11°, and ~17°F, respectively, across the warm tongues (so μ = −6.1, −3.1, and −4.7 K, respectively). From these plots we also estimate (crudely) warm-tongue wavelengths of ~1100 km on 2 and 14 June and ~1300 km on 5 July. From 449-MHz profiler data (along with MPL data on 14 June and 5 July and a 0000 UTC PECAN sounding on 5 July), we estimate 0000 UTC CBL depths at Ellis of ~1.5 km on 2 June, ~2.2 km on 14 June, and ~2.0 km on 5 July. Using these parameters, the -forced analytical model of section 3 produces peak vertical velocities of 4.3, 3.8, and 4.0 cm s⁻¹, respectively. The corresponding peak vertical parcel displacements Δz ~ 725, 605, and 675 m are attained at times t ~ 7.7, 7.1, and 7.5 h, for parcels initially at heights z ~ 650, 1150, and 1050 m.

On 2 June, the first of two rounds of convection initiated around 0900 UTC from a dissipating MCS (Fig. 10d). The northern part of the line and an arc of cells that formed northeast of the MCS may have been affected by the MCS [perhaps triggered by an MCS-generated outflow or bore as in Carbone et al. (1990)], but some of the cells in the southern part of the line appeared to be unrelated to the preexisting convection. About 1100 UTC, a second line of cells with seemingly more pristine origins emerged east of the first line. Imagery from the WV-DIAL, 449-MHz profiler, and MPL at Ellis provided the best evidence of long-lived ascent starting after sunset (Fig. 11). From an estimated lift of 1 km between 0300 and 0800 UTC, we obtain w ~ 6 cm s⁻¹ ( produced by the analytical model was 4.3 cm s⁻¹).

The UFO2 mission on 14 June was described in the mission’s web page notes (http://catalog.eol.ucar.edu/pecan/) as a “surprise CI UFO mission.” Isolated small convective cells developed around 0500 UTC (Fig. 12d) near Hays and over several locations south and east of Hays. The convection gradually intensified and consolidated into three lines. Imagery from the WV-DIAL and 449-MHz profiler at Ellis and (especially) from the CL31 ceilometer at the S-Pol location (Fig. 13) suggests ascent of ~1.5 km from 0200 to 0800 UTC over Ellis and ascent of ~1.5 km from 0400 to 1000 UTC over S-Pol, indicative of w ~ 7 cm s⁻¹ over both locations ( produced in the analytical model was 3.8 cm s⁻¹).

IOP 19 (5 July) provided a good dataset of pristine CI. The first round of convection developed shortly before 0400 UTC in western KS (Fig. 14d). Around 0700 UTC a second line of convection developed east of the first line. The lines drifted eastward and intensified, reaching peak strength between 0900 and 1000 UTC. Imagery from the WV-DIAL and MPL at Ellis (Fig. 15) suggests lift of ~1.3 km from 0500 to 1200 UTC, from which we estimate w ~ 5 cm s⁻¹ ( produced in the analytical model was 4.0 cm s⁻¹). This inferred ascent started about 2 h later than on the other two nights and was actually 1 h after the onset of the first CI.

In a case study of IOP19, Reif and Bluestein (2017) classified the CI on this night as a no-boundary mode. Their figure of the NCEP Global Forecast System (GFS) analysis of the 700-hPa omega (vertical-pressure velocity) field at 0600 UTC showed ascent over much of central and western KS (Fig. 16),^¹¹ though with much mesoscale variability. The peak GFS value () over central KS is similar to the 5 cm s⁻¹ estimate at Ellis.

Trier et al. (2017) examined CI and mesoscale ascent in IOP19 and four other PECAN cases. IOP19 was the only one of their selected cases in which convection initiated without an obvious connection to a front or other surface boundary. Profiles of representative of mesoscale motions were estimated on triangular domains^¹² using PECAN radiosonde winds. For IOP19, the profiles derived from averages of 0300 and 0600 UTC data indicated relatively slow ascent (peak ), which started relatively high above the ground ( for heights above the ~725-hPa level). However, when data at just the later time (0600 UTC) were used, the peak doubled to , and ascent began closer to the ground ( for heights above the ~800-hPa level). These results are consistent with the ascent seen in Fig. 15, that is, low-level ascent of at Ellis that did not begin until ~0500 UTC. Trier et al. (2017) also obtained from 50-member ensemble forecasts initialized at 0000 UTC using data from a mesoscale data assimilation system. The field at 0600 UTC from the ensemble member whose forecast was in closest agreement with observations (Fig. 16a of Trier et al. 2017) was similar to the GFS-analyzed field, at least over KS (Fig. 21d of Reif and Bluestein 2017; our Fig. 16).

5. Summary and final remarks

A hypothesis that weak but persistent mesoscale ascent can arise from the shutdown of turbulent mixing in a horizontally inhomogeneous CBL at sunset is illustrated using a simple 2D model in which the initial state is described by a zero-order jump model of a CBL, and the nocturnal motion is inviscid. The shutdown of turbulence triggers IOs (associated with the horizontal mean part of the CBL) from which Blackadar-like LLJs emerge as well as inertia–gravity waves and persistent (~6–8 h) low-level ascent associated with the inhomogeneity. Two types of inhomogeneity are considered: 1) a laterally varying southerly wind in the CBL and free atmosphere and 2) a lateral variation of the CBL buoyancy, as in a warm tongue. The latter is found to be much more effective than the former in generating vertical motion. For parameter values typical of warm season CBLs, the model yields vertical parcel displacements (e.g., 0.3–1 km) that are of magnitudes that may facilitate nocturnal CI. We note that an extensive north–south-oriented warm tongue is a characteristic feature of the warm season climatology of the Great Plains, as can be inferred, for example, from Fig. 2 of Pu and Dickinson (2014).

For the buoyancy-forced motions we find the following:

The vertical velocity pattern descends with time (more accurately, the zero isoline descends). Low-level ascent is eventually replaced by subsidence.
The peak ascent increases with increasing amplitude of the buoyancy variation, increasing CBL depth, increasing wavenumber k (decreasing wavelength λ) and decreasing N.
The duration of low-level ascent is longer for smaller N and larger λ.
Of the competing tendencies for a smaller λ to increase net vertical parcel displacements by increasing the vertical velocity, but decrease net displacements by shortening the ascent phase, the former is more important.

The ascent produced by the model is in qualitatively good agreement with that inferred in three cases from the PECAN field project. However, because of the assumptions underpinning the model, the subtlety of the physical effects we are exploring, and the uncontrolled nature of field experiments, the comparisons have been necessarily crude. A more rigorous exploration of our theory using controlled experiments with mesoscale numerical weather prediction models should be more illuminating.

Acknowledgments

We thank Dylan Reif, Elizabeth Smith, Petra Klein, and David Parsons (at the School of Meteorology, University of Oklahoma) for helpful discussions about nocturnal CI and LLJs over the Great Plains. William Brown, Scot Loehrer, and Brooks Snyder (at UCAR) provided valuable assistance with PECAN data. This research was supported by the National Science Foundation under Grant AGS-135698.

APPENDIX A

Bessel Functions of the First Kind

We evaluate Bessel functions of the first kind of integer order

(n is an integer,

is real) using standard results described, for example, in AS. For small arguments (

), we evaluate

as the sum of the first 20 terms of the ascending series representation in (9.1.10) in AS (with the gamma function rewritten as a factorial):

For larger arguments (

), we use Hankel’s asymptotic expansion [(9.2.5) in AS] with the first two terms retained in the series [(9.2.9) and (9.2.10) in AS]:

where

Graphs of (e.g., Fig. 9.1 of AS) show that these functions undergo damped oscillations. This behavior can also be seen analytically in (A.2).

It will be convenient to use the recurrence relation in (9.1.27) in AS:

APPENDIX B

Approximating the Forcing Term over a Shallow Slope

For warm tongues in mixed layers similar to those idealized in section 3—but over gentle slopes—the -forcing term in (2.27) can be approximated in terms of along-slope temperature variations. We demonstrate this for slope angles α ~ 0.1° representative of KS (~1-km elevation gain over the ~650-km length of the state), along-slope gradients in of ~1 K (100 km)⁻¹, and <2 K vertical changes in within mixed layers ~1–2 km deep. We work in a slope-following Cartesian coordinate system ( is the downslope coordinate, is the slope-normal coordinate), for which , , and .