ATM 401/501 (Spring 2013):  Daily Class Summary

 

Thu 11 Apr

NOTE:  Exam #2 will be on Tuesday, 16 April.  The exam will cover quasi-geostrophic theory.

 

The in-class severe weather and QPF exercise will begin on Thursday, 18 April.

 

- We resumed and concluded our discussion of severe convective storm forecasting, following the class slides linked below

- Slides 35–97 were discussed, covering:

            - dry slots and cold fronts aloft

            - CAPE/shear phase space and storm mode/structure

            - RKW theory, describing buoyancy/shear interactions near cold pools

            - typical large-scale environments associated with derechos

            - microbursts

            - classic supercell structure

            - splitting supercells

            - case study excerpts from Doswell and Bosart (2001)

             

Tue 9 Apr

- The focus of class was to begin discussion of severe convective storm forecasting, following the class slides (Brief Overview of Severe Storm Forecasting)

- In the most rudimentary sense, moist convection requires lift, instability, moisture, and boundaries (LIMB)

- Slides 1­–34 were discussed, illustrating:

            - the basics of surface fronts and drylines

            - low-level jets

            - vertically coupled front/jet circulations

            - capping inversions and elevated mixed layers

            - characteristic sounding types associated with convective storms

 

Thu 4 Apr

- We continued with our discussion of the Q-vector form of the QG ω-equation following the class notes posted here and here (email Jaymes if you need the ID and password).

- For frictionless, adiabatic, QG flow, the Q-vector form of the ω-equation states that vertical motion is forced by the divergence of the Q-vector.

- In the simplest sense, in order for a Q-vector to be nonzero, we must have (1) a horizontal temperature gradient, and (2) horizontal non-uniformities in the geostrophic wind field.  These horizontal non-uniformities in the geostrophic wind are expressed as relative vorticity and/or horizontal deformation.

- Note that the Q-vector destroys thermal wind balance (see eqs. 13­–16 in the notes).  In doing so, a secondary circulation arises.  This 3-D secondary circulation can be regarded as having vertical branches (i.e., vertical motions) and horizontal branches (i.e., ageostrophic motions).  Both the vertical and horizontal branches are linked by mass continuity.  As a whole, this secondary circulation acts to restore thermal wind balance, to be discussed shortly.

- Where a component of the Q-vector is directed towards warm (cold) air, frontogenesis (frontolysis) by the geostrophic wind is occurring.  Frontogenesis (frontolysis) is associated with a thermally direct (indirect) circulation.  Note that the rising (sinking) branches of these cross-frontal circulations are found where Q-vectors are convergent (divergent).

- According to the scaled horizontal momentum equations (Holton eqs. 2.24 and 2.25), horizontal accelerations (e.g., at jet entrance and exit regions) imply the existence of ageostrophic flow (i.e., departure from geostrophic balance).  Consequently, thermal wind balance is destroyed in these regions.

- Consider an idealized, straight, E-W jet streak atop a uniform N-S temperature gradient.  At the jet entrance region, the horizontal temperature gradient is too large for the vertical shear.  Re-establishing thermal wind balance in this region requires (1) decreasing the horizontal temperature gradient and (2) increasing the vertical shear.  A secondary circulation accomplishes both!

- Note that the Q-vector is directed towards warm air in the jet entrance region, implying geostrophic frontogenesis and a thermally direct secondary circulation.  The vertical branches of this circulation involve rising warm air and sinking cold air, thereby serving to weaken the horizontal temperature gradient, which satisfies requirement (1) above.  The horizontal branches of this secondary circulation involve a southerly ageostrophic wind at jet level and a northerly ageostrophic wind in the low levels.  Owing to the Coriolis force, this ageostrophic flow yields a westerly acceleration at jet level and an easterly acceleration at low levels.  Therefore, the vertical wind shear is increased, which satisfies requirement (2) above.  In this manner, the secondary circulation acts to restore thermal wind balance at the jet entrance region.  A secondary circulation must also exist at the jet exit region, but will be thermally indirect.

- The foregoing interpretation of jet circulations is consistent with the four-quadrant model of idealized jet streaks.  Thus, the circulations associated with jet streaks are predicted by QG theory (and the QG ω-equation).

 

Tue 2 Apr

- We began class by interpreting QG vertical motion from both Eulerian (i.e., fixed-location) and Lagrangian (i.e., flow-following) viewpoints.

- We can undertake a simple thought experiment:  consider an idealized midlatitude shortwave trough in adiabatic, stably stratified, QG flow.  The flow is also equivalent barotropic (i.e., the height contours and isotherms at a given level are parallel), with temperature decreasing in the same direction as geopotential height.   Note that equivalent barotropic flow implies that horizontal temperature advection is zero everywhere.  Furthermore, we will assume that in the upper troposphere, air parcels overtake the trough axis (i.e., flow through it).  In the lower troposphere, air parcels are overtaken by the trough axis.  The Coriolis parameter f is constant everywhere.

           

            Eulerian perspective:  Consider an observer atop a very tall tower located downstream of the trough axis.  As the trough axis approaches, the observer notes a decrease in the temperature.  According to the QG thermodynamic equation (Holton Eq. 6.13), adiabatic cooling via upward vertical motion is the only process that can account for this local cooling, given the conditions stated above.  Therefore, upward (downward) vertical motion is expected ahead of (behind) the trough axis.

 

            Lagrangian perspective:  Consider an observer in the upper troposphere, initially at the trough axis, who is then free to move with the flow.  As the observer proceeds downstream following a particular air parcel, the observer notes a decrease in the relative vorticity of the parcel.  According to the QG vorticity equation (Holton Eq. 6.18), divergence (of the ageostrophic wind) is the only process that can account for this vorticity tendency following the flow, given the conditions stated above.  Therefore, in the upper troposphere, divergence (convergence) is expected ahead of (behind) the trough axis.

            Meanwhile, in the lower troposphere, consider another observer, initially downstream of the trough axis, who is also free to move with the flow.  At this level, the observer is overtaken by the trough axis, and observes an increase in the relative vorticity as the trough approaches.  Again, according to the QG vorticity equation, convergence is the only process that can account for this Lagrangian vorticity tendency.  Therefore, in the lower troposphere, convergence (divergence) is expected ahead of (behind) the trough axis.

            Note that this distribution of divergence and convergence implies upward (downward) vertical motion ahead of (behind) the trough axis.  Furthermore, these vertical motions are wholly consistent with the Eulerian perspective discussed above!

 

- We proceeded with a derivation of the Q-vector form of the QG ω-equation.  The Q-vector was first derived by Hoskins et al. (1978).  See also Sanders and Hoskins (1990).  Note a key conceptual duality:  the Q-vector does to the ω-equation what QGPV does to the height tendency equation.  These two equations, formulated in terms of the Q-vector and QGPV (respectively), are each Galilean invariant and contain single forcing functions (i.e., are not prone to cancellation between terms).

 

Thu 28 Mar

- The main objective of class was to review several forms of the QG ω-equation

- For adiabatic, frictionless, quasi-geostrophic flow, the traditional form of the QG ω-equation includes two RHS terms (or forcing functions): 

            Term A:  differential geostrophic absolute vorticity advection by the geostrophic wind

            Term B:  the horizontal Laplacian of temperature advection by the geostrophic wind

            This equation is discussed in Tom Galarneau’s overview.

- Both terms in the traditional ω-equation can be can be qualitatively evaluated on synoptic charts.  However, there are notable drawbacks to this form of the equation:

            (1)  Terms A and B are not independent.  Note that temperature and geostrophic wind can be written in terms of geopotential (or geopotential height), so that an instantaneous 3-D distribution of geopotential prescribes both terms (A and B).  In other words, if one term is known, then so is the other.

            (2)  Terms A and B (the forcing functions) often oppose each other.  In these situations, qualitative assessment of the total forcing for ω is difficult.

            (3)  Assessment of term A requires information from multiple isobaric levels (i.e., we must evaluate the vertical derivative of geostrophic absolute vorticity advection).

            (4)  Terms A and B are not Galilean invariant, as demonstrated in the class notes.  Note, however, that the sum of terms A and B (i.e., the total forcing) is Galilean invariant.

- We therefore seek to recast the QG ω-equation in a manner that avoids these pitfalls.

- The Sutcliffe-Trenberth form of the ω-equation is once such approach, as derived in the class notes and discussed by Tom Galarneau (see links above).  This equation states that (to a good approximation) ω is forced by a single forcing function: the advection of geostrophic relative vorticity by the thermal wind.  See Trenberth (1978) for the original presentation of this result.  As with classic Sutcliffe Development Theory, note again the significance of the thermal wind in this equation.

 

Tue 26 Mar

- The main objective of class was to introduce quasi-geostrophic potential vorticity (QGPV) and its applications.  QGPV is derived here (as posted on the class website); see also Holton, 4^th edition, sections 6.3.2, 6.3.3, and 6.3.4.

- To motivate development of QGPV, we begin by reexamining the traditional form of the QG height tendency equation (Holton eq. 6.23, which also appears in the derivation linked above).

- Note that in the traditional form of the QG height tendency equation, height tendency is forced by (1) geostrophic absolute vorticity advection by the geostrophic wind, and (2) differential thickness (or temperature) advection by the geostrophic wind.  These forcing functions appear as separate terms on the RHS of the equation.

- At a vorticity maximum or minimum, geostrophic absolute vorticity advection must vanish.  Thus, geostrophic absolute vorticity advection cannot amplify troughs or ridges, but can only move them along.  The amplification mechanism for troughs and ridges is differential thickness (temperature) advection.

- Conceptually, the two terms in the traditional form of the QG height tendency equation are straightforward.  However, in practice, there may be significant cancellation between them.  A cursory inspection of synoptic charts will often identify regions where such cancellation is likely occurring.

- QGPV alleviates this cancellation problem in the height tendency equation.  After deriving QGPV, observe that the height tendency equation may be rewritten in terms of QGPV.  See Tom Galarneau’s discussion here (his section 2).

- According to the QGPV form of the QG height tendency equation, height tendency is forced solely by the advection of QGPV by the geostrophic wind.  Thus, recasting the height tendency equation in terms of QGPV eliminates the cancellation problem, since a single forcing function appears.

- Note that QGPV is simply the sum of geostrophic absolute vorticity and stability.  QGPV is a linearized form of Ertel PV.  For adiabatic conditions, QGPV is conserved following the geostrophic flow on isobaric surfaces (Holton, p. 160).

 

Thu 7 Mar and Thu 14 Mar:

 

NOTE:  Project #1 (macroclimatology) is due on 2 Apr.  Over the next several days, please email a brief synopsis of your project (~1 paragraph) to both Lance and Jaymes.

 

- During these class periods, we completed the derivation of the Sutcliffe Development Equation and discussed its application/interpretation

- The Sutcliffe Development Equation is strictly a diagnostic expression for 1000-hPa convergence.  However, given the relationship between convergence, vorticity tendency, and pressure tendency, we can use this diagnostic expression to make prognostic inferences.

- Note that in all three terms, the thermal wind serves as the advecting flow.  This underscores the fundamental importance of thermal wind balance in understanding midlatitude dynamics.

- Term A may be regarded as a steering term.  This term cannot contribute to surface development, and therefore can only move systems along.  Note again that the thermal wind effectively serves as a steering current for surface cyclones and anticyclones.

- Term B contributes to both development and steering.  Note that assessing term B requires knowledge of the thickness field alone.

- Term C contributes to both development and steering, but is generally small compared to terms A and B.

- We also discussed Laplacian operators and how they may be interpreted:

- The Laplacian of a scalar is simply the divergence of the gradient of that scalar.  Recall that the gradient is a vector field.

- Many fields (e.g., geostrophic relative vorticity) can be written in terms of Laplacians of other fields (e.g., geostrophic relative vorticity is proportional to the Laplacian of geopotential height).  Using geostrophic relative vorticity as an example, this relationship essentially states that the height field directly prescribes the geostrophic relative vorticity field.  In other words, if we have a height field, and wish to know the geostrophic relative vorticity field, then there is no need to compute the geostrophic wind as an intermediate step.

- Remembering that gradients are vectors (having direction and magnitude), we can qualitatively evaluate the divergence of gradients to understand Laplacian operators.

- In the simplest sense, Laplacian operators provide a measure of the distortion of the flow.  In the case of geostrophic relative vorticity, we know that straight, uniformly spaced isohypses yield zero geostrophic relative vorticity.  In such a case, the Laplacian of geopotential height is zero.

 

Tue 5 Mar:

 

NOTE:  Exam #1 will be on Tuesday, 12 March.  The exam will cover forecast verification and ensemble forecasting.  The material presented today will not be on Exam #1.

 

- The focus of class was to discuss the fundamentals of Sutcliffe Development Theory

- Driven by wartime necessity, Sutcliffe sought to find a physical basis for forecasting the intensification/weakening and movement of surface cyclones and anticyclones

- Aircraft limitations of the era meant that upper-tropospheric measurements were not available to forecasters

- Further, Sutcliffe sought to work with spatial derivatives only, which could be qualitatively evaluated on synoptic charts for a single time (i.e., no time derivatives)

- Given these constraints, Sutcliffe appealed to both dynamical and thermodynamical aspects of the problem, and made key simplifications to the governing equations

- Sutcliffe can be credited for inventing QG theory

- Derivation of the Sutcliffe Development Equation is available on the class website, and will be completed in class on Thursday

 

Thu 28 Feb:

- Concluded our discussion of ensemble forecasting topics

- Skilled use of ensemble guidance requires linking “signals” in the ensemble with inferred synoptic and mesoscale features

            - For example, high MSLP spread north of a mean E-W trough suggests high uncertainty in the progression of a warm front

- Often, ensemble mean and spread charts are best used in conjunction with spaghetti charts:

            - Where spread is high, spaghetti charts will help identify whether members tend to cluster around several outcomes, or are uniformly dispersed

- Visualizing ensemble QPF with mean and spread charts is not always useful, especially when the ensemble spread approaches the ensemble mean (often in cases of convective precipitation).

            - Consider QPF decile charts instead

- In addition to simple mean and spread charts, normalized spread is often available

            - For every grid point and lead time, a running mean (e.g., 30 days) of ensemble spread is calculated

            - The spread for the current forecast is then divided by the running mean, thereby normalizing the spread to account for lead time and geography

            - Normalized spread above (below) 1 indicates more (less) uncertainty than the recent average

- See the NCEP ensemble training page and the MetEd COMET module, both discussed in class, for a good review of ensemble forecasting

 

Tue 26 Feb:

- Continued with discussion of ensemble forecasting

- Plotting ensemble mean with ensemble spread highlights the areas of largest forecast uncertainty

- Large spread in the trough access is mostly related to amplitude uncertainty

- Large spread in between troughs and ridges is related to phase uncertainty

- Decile plots show where "X" percent (e.g. 60%) of ensemble members are forecasting at least "Y" inches/mm (e.g. 0.25") of precipitation

- Probability plots illustrate what percentage of ensemble members are forecasting a certain precipitation threshold  

- Plume diagrams display individual ensemble member forecasts of a particular variable (at a given location) as a function of forecast time 

- Precipitation type maps indicate the probability of frozen vs. liquid precipitation 

- Note that we should not overlook the individual ensemble members (especially in the case of large uncertainty)

 

Thu 21 Feb:

- Continued with discussion of ensemble forecasting

- A well-constructed ensemble will occupy a large portion of the actual phase space (i.e., the ensemble will be suitably dispersive)

- Despite having a coarser resolution, the ensemble mean Briar score (forecast skill) exceeds the control run Briar score at all forecast times (on average)

- Ensemble mean offers approximately one additional day of forecast skill

- Features associated with large uncertainty (low predictability) will not be well represented in the ensemble mean.  Instead, the ensemble mean retains the synoptic and global features common to many members, often at lead times of ~6 days.

- Exercise caution when using an ensemble forecast whose members tend to cluster.  This may indicate an underdispersive ensemble.

- Exercise caution when the ensemble spread exceeds the ensemble mean (especially for precipitation forecasts)

 

Tue 19 Feb:

- The focus of class was on ensemble forecasting concepts

- Discussed probability density functions (PDFs) and their relevance to ensemble forecasting

            - A PDF describes the likelihood of observing a certain value (or range of values) for a given field/variable

            - You can create PDFs for many variables (temp, dewpoint, precip, wind speed, etc.)

            - Note that some variables are not normally distributed in the real world (e.g., precipitation amount)

            - A PDF should include the entire range of plausible values (i.e., span the phase space)

            - An underdispersive PDF has an artificially small phase space

            - An overdispersive PDF has an artificially large phase space

            - What are the implied consequences if one solution (or an ensemble of solutions) 

            cannot capture the entire phase space (atmospheric extremes)?

-Discussed probabilistic forecasting vs. deterministic forecasting

            - Deterministic forecasts rely on a single model solution

            - Ensemble forecasts are derived from multiple model solutions

            - Ensemble forecasting allows you to perform statistical analyses of PDFs for a given field (mean, spread, range, etc.)

            - Each ensemble member should be equally likely to verify; however, members from the same parent model tend to cluster near each other 

            - When viewing multiple ensemble solutions, it is important to acknowledge differences in both phase and amplitude (e.g., as illustrated by a 500 hPa height spaghetti diagram)

- Linear climate change involves a shift in the mean (PDF shape is retained)

- Non-linear climate change involves a shift in both the mean and variance (PDF 

      shape changes)

 

Thu 14 Feb:

- Discussed forecast performance of the National Hurricane Center (NHC), Storm Prediction Center (SPC), and Hydrometeorological Prediction Center (HPC).

- NHC:  official storm track errors have improved markedly since 1970, especially at longer time ranges.  Forecasting storm intensity remains a challenge.

- SPC:  many recent severe weather outbreaks have been anticipated several days in advance, which was rare decades ago.  Moderate- and high-risk events are often highlighted with remarkable skill at days 2 and 3.

- HPC:  human forecasters continue to add skill to NWP precipitation forecasts, as measured by threat and bias scores.  Skill tends to be lowest during the summertime, when precipitation is commonly convective in nature.  In 2012, HPC forecasters outperformed MOS in maximum temperature forecasts, but sometimes struggled with minimum temperature forecasts.

 

Tue 12 Feb:

- Discussed the Ocean Prediction Center, which is responsible for maritime weather forecasting

- Major concern is the forecasting of extratropical cyclones producing hurricane force winds

- High frequencies of hurricane force winds overlap with major transoceanic shipping routes

- Implementation of QuickSCAT greatly improved the detection of hurricane force winds, but this instrument is no longer operational

- Many hurricane force wind events go undetected, especially over the Pacific

- Diabatic processes contribute substantially to the explosive deepening of oceanic bomb cyclones

- Intense extratropical cyclones are more common in the SH, where mean meridional baroclinicity is pronounced and expansive

 

Tue 7 Feb:

-  The frequency distributions of GFS and ECMWF ACs for 500 hPa height show that NWP forecast skill has improved remarkably over the past 15 years.  Greatest improvement has occurred in the SH.

-  The same conclusion can be reached by considering the percentage of excellent (AC > 0.9) and poor (AC < 0.7) forecasts for both hemispheres.

-  Root-mean-square error (RMSE) of wind speed in the tropics shows that the GFS is systematically fast compared to the ECMWF.

-  The ECMWF exhibited superior performance in its handling of Sandy, especially at forecast ranges beyond 72 hours

 

Tue 5 Feb:

-  GFS anomaly correlation has increased considerably over the last 20–25 years, but still lags ECMWF (due primarily to differences in data assimilation)

-  Annual mean GFS die-off curves for 500 hPa height AC suggest that useful forecast skill (AC = 0.6) is typically lost at day-8 in the NH, and day-7.6 in the SH.

-  The CDAS (Climate Data Assimilation System) is frozen; any increase in CDAS forecast skill is related to improved data assimilation and initialization

 

Thu 31 Jan:

-  Discussed how terrain influences cyclone movement along the West Coast and the near the Appalachian Mountains.

-  Completed presentation on weather and climate prediction (by A. Thorpe):

-  Sources of predictability (and unpredictability) are found across a broad range of time scales

-  Anomaly correlation (AC, or ACC) is a common metric of NWP forecast skill.  Forecasts verifying with ACs > 0.6 are considered useful. 

-  Advances in NWP capability have yielded a steady increase in ACs over the past three decades for medium range (~ 3–10-day) forecasts.

-  Note especially the shrinking forecast skill gap between northern and southern hemispheres, largely attributable to improved satellite data assimilation practices.

- A fairly comprehensive description of various forecast skill metrics is available

 

Tue 29 Jan:

-  Clarified the mechanism by which a barrier jet can form as a result of terrain-blocked flow.  This is often observed during Appalachian cold air damming episodes.

-  We used simple arguments to deduce that downslope (upslope) flow contributes to cyclonic (anticyclonic) vorticity tendency at the surface.  This relationship explains (1) the development and maintenance of lee troughs and windward ridges, and (2) the synoptic rule-of-thumb that surface cyclones and anticyclones near mountain ranges tend to move with higher terrain on their right (in the NH).

 

Thu 24 Jan:

-  Student-led map discussions will begin on Thu 31 Jan

-  The six questions that map discussions should address are posted on the course website (under Map Discussion).  Prof. Bosart explains the motivation behind this approach in his 2003 paper, Whither the Weather Analysis and Forecasting Process?, available here.