The Planetary Boundary Layer and Uncertainty in Lower Boundary Conditions Joshua Hacker National Center for Atmospheric Research hacker@ucar.edu
Topics The closure problem and physical parameterizations for NWP models. Basic structure and dynamics of the PBL. Some open questions in PBL parameterization and uncertainty in mesoscale forecasting. A column model to emulate a full 3D mesoscale model, and experience with it. Distinction from soil estimation for the sake of soil estimation.
Mesoscales Horizontal wind spectra in the frequency domain. Vinnichenko 1970 Vander Hoven 1957
Reynolds Averaging Key assumption is that Reynolds averaging is valid for numerical weather prediction scales. Viscous eects operate on scales much smaller than energetic eddies. Break variables into mean and turbulent (mean 0) components, where the lter length corresponds to a \spectral gap," or a minimum in energy at a particular range of scales: = + 0 Find equations in the mean to solve, and parameterize the perturbation quantities, including covariance terms.
Closure Prognostic equations for other simplications: @ i @t can be written after averaging and = Advection + body forcings + @u0 j 0 u 0 0 j = f (U; V; W; T; Q; P) body forcings = f (U; V; W; T; Q; P) @x j Subgrid-scale tendencies and body forcings are typically functions of resolved-scale variables and parameters P. Parameters are normally xed values, and can be marginally physical.
Physical Parameterization Any unresolved process that can aect resolved variables.
Physical Parameterization Many cloud processes are below grid scale.
The Planetary Boundary Layer Strong coupling with surface; turbulence. Weak or no coupling to surface; intermittency.
The Planetary Boundary Layer Local Time! Night Day Night
Growth Through Entrainment Heat uxes from the surface generate turbulent eddies that physically mix the air from aloft into the growing PBL.
PBL Depth A dicult problem: how to parameterize the entrainment rate? Typically some function of the surface uxes, with parameters. Conservative scalars (potential temperature, moisture, pollutants) are generally well-mixed within the PBL. Moisture and temperature in the PBL, and inversion strength, are key to thunderstorm prediction.
Soil Moisture Soil moisture partitions the latent and sensible heat uxes through the surface. It can be prognostic or diagnostic.
PBL Analysis and Forecasting Observations in the PBL above the surface are sparse or low-quality. Mesoscale models rely on parameterizations, which are generally decient, to determine the PBL. Mesoscale models are necessary for both the analysis (data assimilation) and forecast problems, when high spatial resolution is required. Better PBL analyses can improve thunderstorm nowcasting, air quality and plume forecasting.
Column Model Environment A 1-D PBL modeling framework: various land-surface and PBL parameterizations, forced. Original model development by Mariusz Pagowski, NOAA/ESRL. Internal dynamics for ageostrophic wind, diusion equation, etc. Geostrophic and radiative forcing from a mesoscale model (e.g. RUC or WRF) or observations. Assimilation of any relevant observations. Here focus on surface obs: T 2, Q 2, U 10, V 10 assimilated half-hourly to get information about the atmospheric state. Cheap! Thousands of realizations possible with a quick turn-around
Model Formulation @U @t @V @t @ @t @Q @t = f (V V g ) = f (U U g ) = = @ @z w0 0 @ @z w0 q 0 @ @z u0 w 0 @ @z v0 w 0 Prognostic in U, V,, and Q with parameterization providing closure. Parameterization is the same as in the Weather Research and Forecast (WRF) model.
A Parameter to Modify Soil Moisture An exchange coecient for moisture, Q c, is computed: Q c = M 1w 0 q 0 q 0 q 1 M is a moisture availability parameter f0,1g. 1 is density at the rst atmospheric model level. q 0 and q 1 are moisture contents at the surface and the rst atmospheric level, repsectively. w 0 q 0 is the parameterized kinematic moisture ux. Provides a lower boundary condition (forcing) for the atmospheric model.
Context (opinions) Numerical weather prediction (NWP) is messy and at times ugly, but its success is undeniable. An honest person will talk about parameter tuning in NWP. Community inertia toward more complex approaches. An alternative may be to seek simpler approaches that make use of objectively tunable parameters. In many cases there is no reason to expect that parameters are stationary in time and space.
Observations to Inform the Model Observations are the only direct source of information about the truth. Observations and model are combined in data assimilation. In data assimilation, observations have the potential to tell us something about the model, including values of parameters.
Linear Statistical Analysis Equation x a = x b + K y o Hx b K = P b H T HP b H T + R 1 x b = x f from the previous forecast cycle (background) H is the forward operator, relating x to y o K weights x b versus y o and describes the covariance between error at the observation sites and error on the model grid P b is the background error covariance matrix (unknown!) R is the observation error covariance matrix, including instrument and representativeness error (unknown!)
Linear Statistical Analysis Equation x a = x b + K y o Hx b K = P b H T HP b H T + R 1 If K is correct, this gives the best linear unbiased estimate of the state x a Estimating P b and R is the crux Ensemble methods provide an approach: estimate P b directly from an ensemble forecast (an ensemble lter)
How an Ensemble Filter Works An Ensemble Filter 6. When all ensemble members for each state variable are updated, have a new analysis. Integrate to time of next observation... y y H H H t k+2 t k * * Model Observation Increment/Update x a = x b + P b H T HP b H T 1 HP b H HP T b H T 1 + R y o Hx b (1)
Skill in PBL State Estimates Height AGL (m) 2500 (a) 2000 1500 1000 500 0 0 1 2 3 4 5 6 7 8 9 MAE ( C) 2500 (c) 2000 Height AGL (m) 2500 (b) 2000 1500 1000 500 0 0 1 2 3 4 5 6 7 8 9 MAE ( C) 2500 (d) 2000 Using only screen-height (surface) observations, skillful proles are estimated at all times of day: (a) 1PM LT, (b) 7PM LT, (c) 1AM LT, and (d) 7AM LT. Height AGL (m) 1500 1000 Height AGL (m) 1500 1000 500 500 0 0 1 2 3 4 5 6 7 8 9 MAE ( C) 0 0 1 2 3 4 5 6 7 8 9 MAE ( C) Figure 2: MAE profiles of for the assimilation experiments (solid) and climatological simulations (dashed), presented in order of increasing time from initialization (0700 LT). Verification times are (a) 1300 LT, (b) 1900 LT, (c) 0100 LT, and (d) 0700 LT.
State Augmentation Data assimilation to estimate a discrete system state Z at time t. Z is a joint state, with both state variables and parameters. X represents state variables. x is a set of parameters, which may or may not be physical. Then Z = (X; x). Given all observations up to the current time, Y t, we want to estimate p(z t jy t ). These experiments are to estimate parameters in a land-surface scheme, given screen-height observations and an evolving model.
Estimate a Single Parameter Spread and error Individual estimates Correlations Forecast Hour Forecast Hour Forecast Hour Single parameters (moisture availability) can be estimated when the true value is known.
Correlations Without Assimilation r( M,T 2 ) r( THC,T 2 ) 0.4 0.2 0 0.2 0.4 0.6 0.8 05/03 05/04 05/05 05/06 05/07 05/08 05/09 05/10 05/11 05/12 05/13 05/14 Analysis Day 0.5 0 0.5 Correlation coecients of T 2 with parameters M and T HC, for 100 ensemble members integrated for 10 days. Parameter distributions are xed. Distributions chosen as with = 0:1M and 0:01T HC. 05/03 05/04 05/05 05/06 05/07 05/08 05/09 05/10 05/11 05/12 05/13 05/14 Analysis Day
Correlations With Assimilation r( M,T 2 ) r( THC,T 2 ) 0.4 0.2 0 0.2 0.4 0.6 0.8 Estimated Not Estimated 05/03 05/04 05/05 05/06 05/07 05/08 05/09 05/10 05/11 05/12 05/13 05/14 Analysis Day 0.5 0 0.5 Estimated Not Estimated 05/03 05/04 05/05 05/06 05/07 05/08 05/09 05/10 05/11 05/12 05/13 05/14 Analysis Day Correlation coecients of T 2 with parameters M and T HC, for 100 ensemble members integrated for 10 days. Parameter distributions are estimated while assimilating. Correlations change, transitions more pronounced.
Dependent Parameters M 0.02 0.01 0.00 0.01 0.02 0.15 0.05 0.05 0.15 0.15 0.05 0.05 0.15 0.02 0.01 0.00 0.01 0.02 0.77 THC M and T HC are linearly dependent when estimated. Here is at 00 UTC for over 10 days, but this is true at any time. Cannot be distinguished, thus could be replaced by a single parameter.
Distribution Improves Assimilation MAE T 2 (K) 3 2.5 2 1.5 1 Fixed, avg = 0.73046 Distributed, avg = 0.26874 Compared to single xed parameter values, distributed parameters result in a better t to observations. The eect is particularly true during transitions. 0.5 0 05/04 05/05 05/06 05/07 Analysis Time (UTC)
Estimation Improves Assimilation MAE T 2 (K) 1.4 1.2 1 0.8 0.6 Estimated, avg = 0.17513 Not Estimated, avg = 0.26874 Compared to xed distributed parameter values, estimated parameters result in a better t to observations. 0.4 0.2 0 05/04 05/05 05/06 05/07 Analysis Time (UTC)
Error in the Prole Dierences in error (estimated { xed distribution) show the prole is generally improved, especially during the growth phase of the PBL.
Summary and Open Questions: Parameter Estimation State augmentation is a useful parameter estimation approach in observation system simulation experiments (OSSEs), but is much more dicult in real-data applications. Much more work to do: How can we deal with non-gaussianity in the parameters? Can we nd distributions that make a better forecast in the face of other, unknown, model errors? Can we nd appropriate stochastic processes to propagate the parameter distributions in time?
Motivating Stochastic Approaches in PBL Forecasting Theoretical: uncertainty in model parameterization is rarely addressed in a meaningful way. Empirical: ubiquitous underdispersion in mesoscale ensemble forecasts suggests that something to approximate model uncertainty is needed. Practical: as a function of the resolved state, parameterizations respond to errors in the resolved state (and thus other parameterization schemes). Process: the response of the PBL to stochasticity, which certainly exists in the real atmosphere, is not well understood.
Example: Radiation at the Surface Incoming solar radiation and outgoing infrared radiation interact with clouds. Do PBL winds, either modeled or in the real atmosphere, react to the modulated radiation in a meaningful way? What temporal and spatial scales of clouds are important?
Daytime Radiative Flux at the Surface Number 1.5 1 0.5 2 x 104 May June 2002 2005 Daytime May-June daytime distribution is bimodal with a broad range of whiteness. This results from shallow convection (boundary layer clouds), and frequent longer-lived thick clouds associated with deep convection. 0 0 200 400 600 800 1000 Direct Normal Shortwave Irradiance at the Surface (W/m 2 )
Model the Eects of Shallow Convection Number 10000 8000 6000 4000 2000 May June 2002 2005 Daytime The eect of the thickest clouds (I < 250 W/m 2 ) removed somewhat arbitrarily. To model the full eects, we need a process to switch between the regimes. 0 200 400 600 800 1000 Direct Normal Shortwave Irradiance at the Surface (W/m 2 )
Normalized Series Normalized Direct Normal Irradiance at the Surface (W/m 2 ) 2 1.5 1 0.5 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 3 3.5 4 Sample x 10 4 Normalized series shows intermittency.
Autocorrelation 1 Autocorrelation 0.9 0.8 0.7 0.6 0.5 Very slow decay with oscillations from lag 250. Weak to moderate autocorrelations are evident. 0.4 0.3 0.2 0.1 0 50 100 150 200 250 300 350 400 450 500 Lag (minutes)
Partial Autocorrelation Partial Autocorrelation 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.9 Statistical signicance is high through about 10 lags. Results seem to suggest that an ARMA model is appropriate for the part of the distribution representing shallow convection. 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 0 5 10 15 20 25 30 35 40 Lag (minutes) 0.1
Next Steps Fit ARMA model, and nd process to switch cloud regimes regimes. Introduce statistical model in column system. Investigate response of the column to variations in the statistical model, in terms of time scale and physical mechanism. What is the expected eect on PBL winds? How can we use this to produce a forecast system that accounts for uncertainty?
Another Potential Focus The stable (nocturnal) boundary layer is characterized by intermittent bursts of turbulence that appear random. Parameterization generally handles laminar ow, and does not account for the intermittency. An extra term could be added to the equations to account for it. Stochastic approaches have been proposed for dealing with the stable boundary layer, but not pursued.
Closing Thoughts The assumptions that go into Reynolds or other averaging techiques, are violated daily in NWP applications. It is time to re-think parameterization; statistical and stochastic approaches are one viable approach. Relatively speaking, deep convection has received the most attention, but the PBL is important at both NWP and climate time scales. I propose a column model to eciently address some of the challanges in PBL parameterization, but caution is necessary to ensure results will extrapolate to the 3D problem.