The intended purpose of the **real-time analyses and forecasts of the quasi-geostrophic (QG) diagnostic equations** web page is to provide an interactive tool that can be used to enhance classroom education and/or weather discussions by providing visualizations of core QG dynamical equations.

The calculations and visualizations are produced using NCAR Command Language version 6.6.2 on computer resources at the NOAA National Severe Storms Laboratory (NSSL). While every attempt is made to keep the images up-to-date, it is possible that there will be interruptions to the image generation services as system updates occur.

In **sections 1-5 **below, all fields are computed using 6-hourly pressure-level data from the National Centers for Environmental Prediction (NCEP) Global Forecast System (GFS) deterministic forecasts available in GRIB2 format at 1×1 degree latitude-longitude grid spacing. The raw NCEP-GFS forecasts are smoothed using a 9-point local smoother run 40 times to produce cleaner results for the QG diagnostics. A detailed description of the fields displayed on each set of images is provided in the Users’ Guides and as captions on the animations themselves. Each animation provided at the links below includes a 10-day analysis archive and the most recent forecast out to 192 hours. Images older than 10 days are not archived. **Current status GFS-based QG diagnostics:** .

In **section 6** below, all fields are computed using the hourly pressure-level data from the NSSL version of the Finite-Volume Cubed-Sphere Dynamical Core (FV3) limited area model run daily at 0000 UTC out to 60 hours. The FV3-NSSL uses the NCEP-GFS for initial and lateral boundary conditions. Physical parameterizations include the NSSL two-moment microphysics and the MYNN PBL as part of the Common Community Physics Package. The FV3-NSSL deterministic forecasts, available at 3 km horizontal grid spacing in GRIB2 format, are interpolated to a 0.1×0.1 degree latitude-longitude grid using patch recovery interpolation prior to computing the mesoscale circulation system diagnostics shown below. A detailed description of the fields displayed on each set of images is provided as captions on the animations themselves. Each animation provided at the links below includes the most recent forecast out to 60 hours. Currently, there is no image archive. **Current status FV3-based mesoscale diagnostics:** . **25 July 2022 status update: Availability of daily FV3-NSSL runs may be intermittent as NOAA Jet HPC undergoes filesystem updates. Apologies for the inconvenience. **

**1. Sutcliffe Development Theory: (click here for Users’ guide)**

- 1000 hPa Z, ζ
_{g}, and 1000–500 hPa ΔZ: NorAmer**||**NH**||**SH - 1000 hPa Z, 1000–500 hPa ΔZ, and ζ
_{T}: NorAmer**||**NH**||**SH - 1000 hPa Z, 1000–500 hPa ΔZ, and total RHS: NorAmer
**||**NH**||**SH - 1000 hPa Z, 1000–500 hPa ΔZ, and steering term (A): NorAmer
**||**NH**||**SH - 1000 hPa Z, 1000–500 hPa ΔZ, and amplification term (B): NorAmer
**||**NH**||**SH - 1000 hPa Z, 1000–500 hPa ΔZ, and Beta term (C): NorAmer
**||**NH**||**SH

**2. Petterssen Development Equation: (click here for Users’ guide)**

- 500 hPa Z and η
_{g}: NorAmer**||**NH**||**SH - 700 hPa Z, T, and ω: NorAmer
**||**NH**||**SH - 1000 hPa Z, 1000–500 hPa ΔZ, and total RHS: NorAmer
**||**NH**||**SH - 1000 hPa Z, 500 hPa Z, and amplification term (A): NorAmer
**||**NH**||**SH - 1000 hPa Z, 1000–500 hPa ΔZ, and steering term (B): NorAmer
**||**NH**||**SH - 1000 hPa Z, 1000–500 hPa ω, and adiabatic term (C): NorAmer
**||**NH**||**SH

**3. Height Tendency Equation: (click here for Users’ guide)**

Traditional Form:

- 500 hPa Z,
**v**and η_{g}_{g}: NorAmer**||**NH**||**SH - 300 hPa Z, T,
**v**, and –_{g}**v**·_{g}**∇**T: NorAmer**||**NH**||**SH - 700 hPa Z, T,
**v**, and –_{g}**v**·_{g}**∇**T: NorAmer**||**NH**||**SH - 500 hPa Z and total RHS: NorAmer
**||**NH**||**SH - 500 hPa Z and propagation term (A): NorAmer
**||**NH**||**SH - 500 hPa Z and amplification term (B): NorAmer
**||**NH**||**SH

QGPV Form:

**4. Omega Equation: (click here for Users’ guide)**

Traditional Form:

- 700 hPa Z and total RHS: NorAmer
**||**NH**||**SH_{ } - 700 hPa Z and -(f
_{0}/σ)[∂(-**v**·_{g}**∇**η_{g})/∂p] term (A): NorAmer**||**NH**||**SH - 700 hPa Z, T, and -(R/σp)[∇
^{2}(-**v**·_{g}**∇**T)] term (B): NorAmer**||**NH**||**SH

Trenberth Form:

- 700 hPa Z, ζ
_{g}, and 900–500 hPa ΔZ: NorAmer**||**NH**||**SH - 700 hPa Z, 900–500 hPa ΔZ, and total RHS: NorAmer
**||**NH**||**SH - 700 hPa Z, 900–500 hPa ΔZ, and (f
_{0}/σ)[2(∂**v**/∂p)·_{g}**∇**ζ_{g}] term (A): NorAmer**||**NH**||**SH - 700 hPa Z, 900–500 hPa ΔZ, and (f
_{0}/σ)[(∂**v**/∂p)·_{g}**∇**f] term (B): NorAmer**||**NH**||**SH - 700 hPa Z, 900–500 hPa ΔZ, and -2D
^{2}(∂θ_{D}/∂p) term (C): NorAmer**||**NH**||**SH

Q- and F-vector Form:

- 700 hPa Z, T,
**Q**, and total RHS: NorAmer**||**NH**||**SH - 700 hPa Z, T, and
**Q**/**Q**/_{n}**Q**intercomparison: NorAmer_{s}**||**NH**||**SH - 700 hPa ψ, T,
**F**, and total RHS: NorAmer**||**NH**||**SH - 700 hPa ψ, T, and
**F**/**F**/_{n}**F**intercomparison: NorAmer_{s}**||**NH**||**SH

**5. QG Energetics: [Lackmann (2011), p. 57–65]**

- 500 hPa Z and Z′: NorAmer
**||**NH**||**SH_{ } - 500 hPa Z and
**v**: NorAmer_{g}′**||**NH**||**SH - 500 hPa Z′,
**v**Φ′, and -∇·(_{a}′**v**Φ′): NorAmer_{a}′**||**NH**||**SH

**6. Experimental Mesoscale Circulation System Diagnostics:**

This section extends QG diagnosis of vertical motion to the mesoscale in the alternative balance framework by replacing the geostrophic wind with the nondivergent wind. The goal is to test the utility of the balanced framework in producing clean diagnostic signatures of mesoscale ascent and frontogenesis in convection-allowing models (CAMs). This work is motivated by Nielsen-Gammon and Gold (2008), Kenyon et al. (2020), and discussions with Dan Keyser. The plots below are derived from the once-daily NSSL FV3-SAR initialized at 0000 UTC and run out to 60 hours. Additional visualizations are available at the NSSL CAMs website.

The diagnostic plots below will require additional tuning to attain a proper balance between spatial smoothing and representation of coherent mesoscale features. The plots represent work that is in progress and should not be considered a final product.

CONUS-scale overview plots:

- T
_{b}, 250 hPa relative vorticity, wind speed, and vector divergent wind - SLP, 1000-500 hPa thickness, and 250 hPa wind speed
- 500 hPa Z, T, relative vorticity, and vector wind
- Total column PW, 700 hPa Z, T, and vector wind
- SLP, 850 hPa T, 10-m vector wind, and 1-h precipitation

Regional-scale vorticity plots: (**v _{g}** and

**v**intercomparison)

_{nd}- 250 hPa ζ
_{g}/**v**and ζ/_{g}**v**: NW || SW || NP || SP || NE || SE_{nd} - 500 hPa ζ
_{g}/**v**and ζ/_{g}**v**: NW || SW || NP || SP || NE || SE_{nd} - 700 hPa ζ
_{g}/**v**and ζ/_{g}**v**: NW || SW || NP || SP || NE || SE_{nd} - 850 hPa ζ
_{g}/**v**and ζ/_{g}**v**: NW || SW || NP || SP || NE || SE_{nd}

Regional-scale vector frontogenesis (**F)** plots:

- 700 hPa θ,
**F**, -2**∇**·**F**: NW || SW || NP || SP || NE || SE - 850 hPa θ,
**F**, -2**∇**·**F**: NW || SW || NP || SP || NE || SE

Regional-scale frontogenesis plots:

- 700 hPa Petterssen frontogenesis: NW || SW || NP || SP || NE || SE
- 850 hPa Petterssen frontogenesis: NW || SW || NP || SP || NE || SE

CONUS-scale upscale feedback plots:

**7. References:**

Bluestein, H. B., 1992: *Principles of Kinematics and Dynamics*. Vol. I. *Synoptic-Dynamic Meteorology in Midlatitudes*. Oxford University Press, 431 pp.

Carlson, T. N., 1998: *Mid-Latitude Weather Systems.* Amer. Meteor. Soc., 507 pp.

Davies-Jones, R., 1991: The frontogenetical forcing of secondary circulations. Part I: The duality and generalization of the Q vector. *J. Atmos. Sci.,* **48,** 497-509.

Hakim, G. J., L. F. Bosart, and D. Keyser, 1995: The Ohio Valley wave-merger cyclogenesis event of 25-26 January 1978. Part I: Multiscale case study. *Mon. Wea. Rev.,* **123,** 2663-2692.

Hoskins, B. J., I. Draghici, and H. C. Davies, 1978: A new look at the omega equation. *Quart. J. Roy. Meteor. Soc.,* **104,** 31-38.

Kenyon, J. S., D. Keyser, L. F. Bosart, and M. S. Evans, 2020: The motion of mesoscale snowbands in Northeast U.S. winter storms. *Wea. Forecasting,* **35,** 83-105.

Keyser, D., M. J. Reeder, and R. J. Reed, 1988: A generalization of Petterssen’s frontogenesis function and its relation to the forcing for vertical motion. *Mon. Wea. Rev.,* **116,** 762-780.

Keyser, D., B. D. Schmidt, and D. G. Duffy, 1992: Quasigeostrophic vertical motions diagnosed from along- and cross-isentrope components of the Q vector. *Mon. Wea. Rev.,* **120,** 731-741.

Lackmann, G., 2011: *Midlatitude Synoptic Meteorology: Dynamics, Analysis, and Forecasting.* Amer. Meteor. Soc., 345 pp.

Martin, J. E., 1999a: Quasigeostrophic forcing of ascent in the occluded sector of cyclones and the Trowal airstream. *Mon. Wea. Rev.,* **127,** 70-88.

Martin, J. E., 1999b: The separate roles of geostrophic vorticity and deformation in the midlatitude occlusion process. *Mon. Wea. Rev.,* **127,** 2404-2418.

Martin, J. E., 2006: *Mid-Latitude Atmospheric Dynamics: A First Course*. John Wiley & Sons Ltd., 324 pp.

Nielsen-Gammon, J. W., and D. A. Gold, 2008: Dynamical diagnosis: A comparison of quasigeostrophy and Ertel potential vorticity. *Synoptic-Dynamic Meteorology and Weather Analysis and Forecasting: A Tribute to Fred Sanders, Meteor. Monogr.,* No. 55, Amer. Meteor. Soc., 183-202.

Petterssen, S., 1956: *Motion and Motion Systems.* Vol. I. *Weather Analysis and Forecasting.* McGraw-Hill, 428 pp.

Sutcliffe, R. C., 1947: A contribution to the problem of development. *Quart. J. Roy. Meteor. Soc.,* **73,** 370-383.

Sutcliffe, R. C., and A. G. Forsdyke, 1950: The theory and use of upper air thickness patterns in forecasting. *Quart. J. Roy. Meteor. Soc.,* **76,** 189-217.

Trenberth, K. E., 1978: On the interpretation of the diagnostic quasi-geostrophic omega equation. *Mon. Wea. Rev.,* **106,** 131-137.