Parameterizing Richardson number hysteresis in the boundary layer

Ron McTaggart-Cowan and Ayrton Zadra

World Weather Open Science Conference

20 August 2014

PBL Background Hysteresis SCM DiscussionFreezing Rain DA Cycle

Outline

• Representing the planetary boundary layer (PBL) in NWP

• Turbulence regime transitions and Richardson number (Ri) hysteresis

• Impact of Ri hysteresis:– Column model experiment– Freezing rain case study– Data assimilation cycles

• Synthesis and discussion

PBL Background

PBL Background Hysteresis SCM DiscussionFreezing Rain DA Cycle

Representing the PBL

• The PBL is defined as the lower portion of the atmosphere in which the atmosphere feels the effect of the surface on hourly timescales through turbulent eddies (Stull 1988)

• In numerical models, transports of heat, moisture and momentum by these eddies are usually represented by enhanced vertical diffusion

• The strength of the unresolved eddies must be estimated based on resolved quantities using a closure assumption

• In operational NWP, first-order closures are standard, in which turbulence is related directly to state variables in the model

PBL Background

PBL Background Hysteresis SCM DiscussionFreezing Rain DA Cycle

Representing the PBL

Where,

Shear generation orbuoyant suppression Viscous dissipation Vertical diffusion

λ - Mixing lengthc – ConstantK – Diffusion coefficient

Where, is the Prandtl number.

Buoyant Suppression

Shear Generation

PBL Background

• The Global Environmental Multiscale (GEM) model run by the Canadian Meterological Centre is an exception, using a 1.5 order closure based on predicted turbulent kinetic energy (TKE, E)

• The sign of the first term dominates the overall TKE tendency, determined by B

• The Richardson number (Ri) thereby largely controls the growth/decay of TKE

PBL Background Hysteresis SCM DiscussionFreezing Rain DA Cycle

Representing the PBL

PBL Background

• For Pr=1 (Taylor 1914), this Ri-B relationship has a “critical” value at Ri

c=1

• Physically, this Ric is the point at which the flow

transitions from turbulent (Ri<Ric) to laminar (Ri>Ri

c)

• Both linear theory (Taylor 1931) and observations (Oke 1970) suggest much lower values for Ri

c: 0.25

and ~0.1, repsectively

• Time-dependency in the background and the influence of Pr≠1 may explain some of these differences (Majda and Shefter 1998), but a closer look at the transition is needed for TKE-based closure

PBL Background Hysteresis SCM DiscussionFreezing Rain DA Cycle

Richardson Number Hysteresis

• In a flourescent dye experiment at the thermocline, Woods (1969) found that tubulence regimes persist across a subregion of Ri space 0.25 < Ri < 1

– Turbulent flow remains turbulent until Ri > 1– Laminar flow remains laminar until Ri < 0.25

• Through a TKE budget, Businger (1969) attributed this asymmetry to the fact that energy generated in the transition zone is available only to finite-amplitude perturbations associated with pre-existing turbulence

• This regime persistence is a classic example of hysteresis, a common concept in electromagnetics and engineering: it is used in thermostats to avoid rapid switching around the set temperature

Hysteresis

PBL Background Hysteresis SCM DiscussionFreezing Rain DA Cycle

Richardson Number Hysteresis

• The current regime (laminar or turbulent) can no longer be diagnosed directly from Ri, since information about the previous regime is needed

• This leads to a modified Ri-B relationship with regime-dependent branches

Relationship between the TKE growth parameter (B) and Ri for the original PBL scheme (CTL, left) and a double loop (HYST2, right). The turbulence regime is identified by line style as shown in the CTL legend, and the direction of travel on each branch is shown by arrows for HYST2. A normalization factor B

o=B(Ri=0) is used on the

ordinate.

Hysteresis

PBL Background Hysteresis SCM DiscussionFreezing Rain DA Cycle

Single Column Model (SCM) Results

• The GABLS-3 intercomparison case is used to assess the impact of Ri hysteresis during diurnal transitions

Local midnight profiles from the 12-h SCM forecast of the GABLS-3 case (Bosveld and Coauthors 2014) from the CTL (black) and HYST2 (grey) PBL formulations.

The nocturnal inversion is deeper and stronger in HYST2, with low Ri values at the nose.

A stronger low level jet is supported in HYST2, since shear producing Ri<1 does not immediately lead to turbulent mixing.

SCM

PBL Background Hysteresis SCM DiscussionFreezing Rain DA Cycle

Single Column Model (SCM) Results

• The impact of Ri hysteresis on the timing and duration of turbulence in the PBL can also be assessed in the simplified SCM context

Time series of TKE at ~450 m in the GABLS-3 SCM case, beginning at 1 pm local time. Periods during which the flow in each integration (CTL, black; HYST2, grey) is laminar are idenified at the bottom of the plot.

Both the evening and morning transitions are slightly delayed in the HYST2 integration.

Similarly, the pulses of intermittent mixing overnight are delayed, last longer, and decay more slowly when Ri hysteresis is active (HYST2).

Both delay and damping are expected consequences of the hysteretic loop.

SCM

PBL Background Hysteresis SCM DiscussionFreezing Rain DA Cycle

Freezing Rain Case Study

• A common problem with PBL schemes is over-mixing during warm advection by a low-level jet, an error mode known termed warm episode by EC forecasters

• Screen-level temperature guidance errors can exceed 5oC in day-1 predictions during warm episodes

• The strong southwesterly flow in the warm sector of a winter continental cyclone is a preferred region for the occurrence of this error mode

• These errors have a particularly large impact on freezing rain forecasts because of the sensitivity of this precipitation type to the inversion profile

• A canonical warm episode case initialized at 0000 UTC 22 March 2007 is investigated using the GEM global configuration with the operational 0.3o grid

Freezing Rain

PBL Background Hysteresis SCM DiscussionFreezing Rain DA Cycle

Freezing Rain Case Study

Analyses from ERA-Interim for 0000 UTC and 1200 UTC 22 March. Panels show: (a,b) dynamic tropopause potential temperature and winds, and 925-850 mb relative vorticity; (c,d) 850 mb temperature and winds; (e,f) 2-m temperature and sea level pressure. Fronts from HPC are plotted in c-f, and winds are plotted in knots.

An upper level trough begins to tilt negatively as the associated developing cyclone near James Bay creates a cold front over the midwest, with strong warm advection across the Great Lakes and US northeast.

The warm front splits along the snow line, creating a large region over southeastern Canada and northeastern US with near-freezing surface air temperatures.

Freezing Rain

PBL Background Hysteresis SCM DiscussionFreezing Rain DA Cycle

Freezing Rain Case Study

Errors in the 12-h guidance for screen-level temperature with respect to ERA-Interim (colour-shaded) and observations (colour-coded dots). Only observations and analysis grid cells with surface elevations that lie within 100 m of the GEM model orography are plotted. The maximum warm episode region (errors >4 K) is shown with a heavy solid line.

The warm episode error mode is evident in this case, with 12-h screen level guidance errors in excess of 4oC that cover much of eastern Ontario and southwestern Quebec.

These near-surface temperature errors are the result of excessive downwards mixing of warm air from the nose of the inversion, which is maintained by the strong 850 mb warm advection.

Instead, shear below the low-level jet in CTL leads to Ri<1, TKE growth and large vertical diffusion.

Freezing Rain

PBL Background Hysteresis SCM DiscussionFreezing Rain DA Cycle

Freezing Rain Case Study

The potential freezing rain region during the period is shown on the left. Above are ERA-Interim and CTL profiles at 1200 UTC within the region, with temperature differences shown on the right.

Excessive PBL mixing leads to a weakened warm nose compared to ERA-Interim and observations: EC warnings are issued only after freezing rain begins.

Freezing Rain

PBL Background Hysteresis SCM DiscussionFreezing Rain DA Cycle

Freezing Rain Case Study

Errors in the 12-h guidance for screen level temperature are shown on the left for CTL (top) and HYST2 (bottom). Profile differences from ERA-Interim in the potential freezing rain region are plotted above for CTL (left) and HYST2 (right).

The impact of Ri hysteresis is consistent with the SCM results: turbulent vertical mixing is reduced and the nose of the inversion remains strong.

CTL HYST2Near-surface temperature guidance is improved by ~2oC in the warm error region, without degradation elsewhere.

Complete correction of the warm episode is impossible because errors have accumulated in the analysis cycle.

Freezing Rain

PBL Background Hysteresis SCM DiscussionFreezing Rain DA Cycle

Freezing Rain Case Study

Relevant PBL quantities after 6-h of integration in CTL (left) and HYST2 (right). Top row shows TKE (shaded as indicated on the greyscale bar) and mixing length, while bottom row shows the vertical diffusion coefficient for heat (K

H, shaded as indicated on the greyscale bar).

The broad region of enhanced TKE between the split warm fronts, combined with large mixing lengths, is dramatically reduced with Ri hysteresis.

These factors lead directly to a reduction in the estimate of turbulent vertical diffusion through reduced transfer coefficients.

The PBL scheme is less active in this region through-out the HYST2 integration.

TKE

KH

Freezing Rain

PBL Background Hysteresis SCM DiscussionFreezing Rain DA Cycle

Freezing Rain Case Study

Accumulated freezing precipitation (shaded as shown on the greyscale bar) over the first 12-h of integration in CTL (top) and HYST2 (bottom). Stations reporting at least one hour of freezing rain are shown with the freezing precipitation symbol, while those reporting only non-freezing precipitation are shown with a dot. A total of 53 stations reported precipitation type with in the plotted region over this time period.

The impact of including the effect of Ri hysteresis extends into precipitation type guidance because of the improved lower-tropospheric temperature profile.

The region covered by freezing rain predictions extends to match observations more closely, with the equitable threat score for 0.5 mm accumulations rising from 0.2 (CTL) to 0.35 (HYST2), a combination of increased probability of detection and decreased false alarm ratio.

Freezing Rain

PBL Background Hysteresis SCM DiscussionFreezing Rain DA Cycle

Data Assimilation Cycle

• The case study framework allows us to develop an understanding of the physical impact of Ri hysteresis on a potentially high-impact event, however it is inherently limited:

– Is the identified impact representative of performance across many cases?

– What effect does the unmodified analysis have on the differences between CTL and HYST2?

• A full data assimilation cycle is needed to address these questions

• A 4D-Var cycle for the February-March 2011 period* is run using both CTL and HYST2 PBL formulations, with a 0.25o grid spacing

* The results of a 2-month summer cycle are not shown because the dominance of convective PBLs limits the impact of Ri hysteresis.

DA Cycle

PBL Background Hysteresis SCM DiscussionFreezing Rain DA Cycle

Data Assimilation Cycle

Results from 24-h forecasts run from the CTL (black) and HYST2 (grey) data assimilation cycles, shown as bias (dotted) and standard deviation (solid) with respect to Northern Hemisphere radiosondes, with the number of observations shown on the right. Differences between the curves are plotted in the right-hand panels, with negative values indicative of error reductions and the 99% confidence interval shaded.

The introduction of Ri hysteresis in the assimilation cycle leads directly to improved short-range temperature, height and wind forecasts in the PBL.

Improvements at mid-levels and at longer lead times (120 h forecasts not shown) were unexpected, but appear to be robust.

The impact of Ri hysteresis found in the case study therefore appears to be representative and remarkably positive in the cycle.

DA Cycle

PBL Background Hysteresis SCM DiscussionFreezing Rain DA Cycle

Discussion

• The use of a TKE closure in PBL schemes requires a knowledge of the turbulence state to determine TKE growth or decay; however, transition does not appear to occur at a singular Ri

c

• The introduction of an observationally based hysteretic loop in the Ri-B relationship reduces the tendency for over-diffusion of the warm nose in the presence of warm advection by a low-level jet

• Alternative formulations (intermittent turbulence, total energy) may have similar impacts on forecast profiles, and are under development

• The success of Ri hysteresis in reducing warm episode errors and improving guidance quality has led to its implementation in all operational systems

Discussion

PBL Background Hysteresis SCM DiscussionFreezing Rain DA Cycle

References

Bosveld, F. and Coauthors, 2014: The third GABLS intercomparison case for model evaluation. Boundary Layer Meteorology, doi:10.1007/s10546-014-9919-1.

Businger, J. A., 1969: Note on the critical Richardon number(s). Quart. J. Roy. Meteor. Soc., 95, 653-654.

Majda, A. and M. Shefter, 1998: Elementary stratified flows with instability at larger Richardson number. J. Fluid Mech., 376, 319-350.

Oke, T. R., 1970: Turbulent transport near the ground in stable conditions. J. Appl. Meteor., 9, 778-786.

Stull, R. B., 1988: An introduction to boundary layer meteorology. Klewer Academic Publishers, 670 pp.

Taylor, G. I., 1914: Eddy motion in the atmosphere. Phil. Trans., 215, 1-26.

Taylor, G. I., 1931: Effect of variation in density on the stability of superposed streams of fluids. Proc. Roy. Soc. London A, 132, 499-523.

Woods, J., 1969: On Richardson’s number as a criterion for laminar-turbulen-laminar transition in the ocean and atmosphere. Radio Science, 4, 1289-1298.

Discussion