The Balancing Act of Geostrophic Adjustment

1.0 Introduction

The maintenance of real and model weather systems and the model's response to new data can be anticipated and understood with the help of geostrophic adjustment theory, which describes how a fluid responds to a perturbation imposed upon it.

For instance, consider a mesoscale convective complex (MCC) (MCC is only an example; it could be any sort of disturbance). The MCC blows up dramatically and quickly, injecting low-level air through the depth of the troposphere, forcing the ambient air out of the way, and generating substantial wind and temperature perturbations over some region. Once the MCC has completed its life cycle, it leaves behind an upper-tropospheric outflow jetlet, a mid-level vortex, and a low-level boundary. The material presented here should help for three types of forecast puzzles that may occur:

1. Real weather: The MCC occurs. How long will its residual features last? How large an area will come under their influence? And how will their mass and wind fields evolve?

2. Model forecast: The model makes its version of an MCC with attendant residual features, but it appears that no MCC will actually develop. How long will these features last? How large an area will come under their influence? And how will the model evolve their mass and wind fields?

3. Data assimilation: The MCC occurs, but the model does not predict it. However, new data getting into the next model run has sampled the environment changes. What is the impact of these data on the new model forecast?

How the atmosphere responds to a disturbance imposed upon it is a well-studied problem in classical geophysical fluid dynamics. It has direct application to the real atmosphere and the model atmosphere, and it forms the basis for understanding how new data affect the model forecast. The impact of new data on the analysis comes in the form of analysis increments, which are changes from a first-guess forecast. Then, the impact of the new data on the forecast depends on how the model responds to the analysis increments.

Unfortunately, geostrophic adjustment is often taught only in graduate-level dynamics classes, although the consequences are of fundamental practical value in understanding the behavior of both the real atmosphere and the models. The basic concepts and their application will be summarized here. The theoretical development, even on a basic level, will be skipped. Some mathematical development from first principles can be found in textbooks such as Haltiner and Williams (1980) and Gill (1982) and the original references therein, and searching the Web on "geostrophic adjustment" will yield many journal articles.

The explanations here are broken up into six pages. For it to make sense, it is essential to complete the first three pages in sequence, as they provide the underlying conceptual background material.

Basic references

Gill, A. E., 1982: Atmosphere-Ocean Dynamics. Academic Press, New York, 662 pp.

Haltiner, G. J. and R. T. Williams, 1980: Numerical Prediction and Dynamic Meteorology. (Second Edition) John Wiley & Sons, New York, 477 pp.

References on specific dynamics issues not covered in these texts. For instance, the effect of relative vorticity on the inertial period are found in various journal articles spanning several decades.

Unfortunately, it seems that all relevant references are of a theoretical rather than practical/applied nature and most are designed for research and advanced graduate-level work in dynamics and data assimilation.

2.0 Scenario

To start with, suppose the winds and heights are in perfect geostrophic balance when a disturbance is suddenly imposed upon it. The atmosphere responds by sending out gravity wave pulses, spreading out like ripples on a pond when a rock is dropped. In the wake of these gravity waves is left behind a new geostrophically balanced state. The new state is different from the original geostrophic conditions in both the winds and heights. These changes remain for a long time because the new state is balanced. This process of the atmosphere evolving toward a balanced state is called "geostrophic adjustment."

This schematic shows the sequence of events caused by suddenly imposing a height depression with no wind perturbation on a straight, geostrophically balanced, large-scale flow. The undisturbed geostrophic conditions are given by the black height and wind curves. The height (wind) perturbations associated with the disturbance at each time are shown with red (blue) shading. Disturbed states at the "initial" and "later" times are also overlayed on the "later" and "final" times for comparison.

The spreading influence of the disturbance is heralded by the first wave propagating outward (the wave that has reached the furthest away from the disturbance center). Ahead of the wave, conditions remain undisturbed until the wave arrives, while conditions in the area where all the gravity have finished passing will have reached their final state. Eventually, smooth new geostrophically-balanced mass and wind fields result, with distant locations unaffected by the disturbance.

The final state also would include inertial oscillations (like winds in the upper part of the boundary layer after nocturnal decoupling), but these are unimportant to the concepts and applications discussed here. The disturbance could be a convectively-generated vorticity maximum or a short wave or a cyclone that didn't appear in the model (or one in the model that doesn't appear in any observations). Or it could be whatever you want to partition off as "disturbance." For considering the influence of new data on the model forecast, the disturbance is the field of analysis increments resulting from that new data getting into the analysis.

In reality and in the model, the initial balanced state is not geostrophic; it includes curved flow, frontal circulations, jet streaks, and divergence associated with latent heating. Most of the time the mass and wind fields are closely coupled, slowly evolving in a "quasi-balanced" manner.

The changes to these balanced winds and heights are the response to the disturbance. For data assimilation, the response to the new data is the change in the forecast. The forecast includes the gravity wave ripples, but they go away early in the forecast period. If the disturbance grows gradually rather than being thrown in all at once, the gravity-wave part of the response is diminished but similar long-lived adjustments in the winds and heights will result.

Matters of greatest practical importance to determine are:

3.0 Size Matters!

The key to the response is whether the disturbance is much wider, comparable to, or much less than the Rossby radius of deformation. The Rossby radius is related to the distance a gravity wave propagates before the Coriolis effect becomes important. Essentially, the gravity waves are able to disperse across this distance, as seen by the wiggly height pattern in the middle frame of the animation on the previous page. The gravity waves leave behind an adjusted state, as seen on the last frame, but they are unable to disperse across a broader distance, leaving areas beyond this distance unchanged.

The disturbance wavelength (crest-to-crest) L gets compared to .

This (2L_R) is the distance the gravity wave could travel during an inertial period, which, in turn, is the amount of time it would take for an ageostrophic wind vector to completely pivot around a circle as a result of being turned by the Coriolis effect.

For disturbances of intermediate scale, the result is between these extremes.

4.0 How large is "large scale?" Depth, stability, and vorticity matter

How large is 2L_R? That depends primarily on the vertical depth of the disturbance and to a lesser degree on the lapse rate and absolute vorticity. The first two of these factors enter the picture because they affect the gravity wave speed. The third enters because it affects the inertial time scale. Also, 2L_R is much larger in the tropics because the small Coriolis parameter there makes the inertial period much longer than in midlatitudes.

2L_R  =  gravity wave speed  x  inertial period        

2L_R  =  gravity wave speed  x  inertial period

How long is the inertial period?

Practical impact

The classic expression for the inertial period, 2/f, only applies in the absence of a horizontal pressure gradient. In uniform horizontal shear, substitutes for the Coriolis parameter, f, where is the absolute vorticity following a parcel (such as on an isentropic surface in unsaturated conditions). For a vortex in solid body rotation, the Rossby radius just uses . There is no general formula for more complicated flows, but a generally reasonable approach is to go with

Because the absolute vorticity and Coriolis parameter are in the denominator, The Rossby radius is smaller for a cyclone and larger for an anticyclone, typically by a factor of around two compared to average conditions The Rossby radius is larger at lower latitudes, with 20° latitude a factor of two larger than at 45° latitude when the relative vorticity is very small

Putting these pieces together, we find the critical length scale for a disturbance is

which comes out to this most usable form in units of kilometers

where

N	is the Brunt-Vaisala frequency (~.008 s-1 for steep lapse rates of 8 K/km, ~.02 s-1 for isothermal conditions)
H	is the disturbance depth in km
fo	is 10 x 10-5 s -1 (10 "units" on your vorticity map)
f	is the Coriolis parameter (6 x 10-5 s-1 at 25°N, 11 x 10-5 s-1 at 50°N)
	is the absolute vorticity

Thus, whether a feature is dynamically "large-scale" or "small-scale" depends on its stability, its depth, and local and planetary contributions of vorticity.

Remember, this geostrophic adjustment process is in addition to other forcing. As long as a feature is being forced, it will continue to exist, with its structure evolving as determined by the forcing and gradual modification by the adjustment process. Also remember that vertical shear and horizontal deformation can shred a feature over time, even if the dynamics permit a thermal or wind perturbation to otherwise be retained. This is just one more tool in your bag, not the answer to all forecast problems.

5.0 Application to Real Weather Systems

Big picture

The Rossby radius in the tropics is very large. Consequently, most mesoscale features are short-lived and the wind field is more important than the mass field in determining their future evolution. Monsoon circulations and planetary-scale features such as the Madden-Julian Oscillation (MJO) are large enough to have longevity and for the mass field to be more important in the forecast. Hurricanes have sufficiently high vorticity to greatly reduce the Rossby radius, enabling them to survive for days instead of hours.

The Rossby radius in midlatitudes is smaller than the longwave ridges and troughs, which are controlled by the mass field. Mobile shortwave troughs are a mixture, depending on their strength and wavelength. Mass and wind information are both important in determining the evolution of shortwave troughs. Features in an anticyclonic environment have to be larger than features in a cyclonic environment to have longevity.

Real-world differences from theory

The real atmosphere usually has fairly smooth evolution rather than a sudden imposition of disturbances followed by wave pulses and an adjustment process. However, potent, rapidly developing systems sometimes do undergo fairly sudden changes in which gravity waves may play a role in an adjustment process.

Gravity waves are ubiquitous in the real atmosphere — caused by fronts, cumulus clouds, flow over hills as well as mountains, cold air drainage, and many other things. Typically they are of small amplitude and short horizontal wavelength, propagating their energy vertically rather than spreading energy over large horizontal distances. They do not accomplish much (if any) geostrophic adjustment.

The theory is based on barotropic conditions — no phase tilt or change in perturbation sign with height. A baroclinic wave has a phase tilt while a hurricane has an anticyclone sitting on top of the cyclone. These kinds of complications make application sometimes less than straightforward.

Nonetheless, the essence of the theory — the longevity of a feature and whether it is controlled primarily by the wind field or the mass field — does hold up. The atmosphere is always being thrown out of balance — by mountains, convection, turbulence, jet streaks accelerating parcels through regions of near zero absolute vorticity, movement of sharp boundaries created by geography (like elevated mixed-layer plumes), etc. Usually the forcing throwing things out of balance gradually ramps up and secondary, somewhat lagged, circulations form in response. Thus, the atmosphere can be thought of as always undergoing an adjustment.

Practical forecast application

Suppose an event occurs, creating some disturbance, and then its forcing dissipates. It might be an MCC mid-level vortex and upper-level outflow jet, it might be a cutoff 500hPa cold pocket, or perhaps a warm, elevated, mixed-layer plume. To apply this concept in your forecast process, follow these three steps:

1) Consider the feature as a perturbation on the larger-scale background conditions
2) Estimate its characteristic depth, width, stability, and average vorticity
3) Determine the Rossby radius — is the feature longer than 2L_R?

The constant gentle pitter-patter of short wavelength gravity waves in the real atmosphere can be reproduced in models with grid spacings of 1 or 2 km but becomes more like silence interrupted by a periodic "boom!" in models with resolutions typically used for NWP. The nature of the adjustment process in response to imbalances in initial conditions and imbalances created by forcing from latent heating and forcing from other model physics is often impulsive. Thus, model behavior corresponds more closely to the theory described on the first few pages here than to the smoother adjustment process typical in the real atmosphere.

What do we mean by "balanced" mass and winds?

Model fields are "balanced" when the dynamics and physics in the model are consistent.

• Divergence and vertical motion are consistent with forcing caused by surface friction, latent heating, flow over orography, frontal upglide, jet streak dynamics, etc.
  

• Changes in divergence and vertical motion following a weather feature tend to be gradual
  

Conditions may

• Start out unbalanced because the analysis may not be entirely consistent with the model topography, physics, resolution, or other aspects of the model
  

• Become unbalanced during the forecast due to sudden bursts of latent heating by convection, by fast flow over topography, or any other quickly-imposed strong forcing

When the model has unbalanced conditions, it emits gravity waves and adjusts toward a balanced state. The intensity, scale, and duration of the impact on the forecast depends on the magnitude of the imbalance and whether it is dynamically large or small (remember, its wavelength, L, compared to 2L_R).

How initial imbalance affects the forecast

Unbalanced initial conditions cause gravity wave noise early in the forecast. The adjustment process may also change the large-scale features somewhat. Here is an example from the 22-km operational Eta model and the nested 10-km Eta regional run.

The 22-km Eta fields are smooth. The 10-km Eta isobars are wiggly pretty much all over, with even some isolated spots of 1032mb over central AR and northern LA.

• The wiggles are characteristic of running a model without balanced initial conditions

• The wiggles in this case may also be partly due to latent heating in the model cyclone (discussed in the next section below)

• 3D-VAR and assimilation cycling combine to make smooth, "balanced" initial conditions for the 22-km Eta model

• The wiggly behavior typically results from using model initial conditions purely from data or from a different model or from the same model at different resolution, because of the inconsistencies introduced

• Initialization procedures modifying model analysis fields are used to minimize this problem when the analysis procedure does not generally yield sufficiently balanced initial conditions

• The 10-km Eta initial conditions were interpolated from the 22-km Eta, and a digital filter was applied to eliminate small-scale noise. Testing in other cases without using the filter showed considerably worse problems
  

Also, notice the larger-scale modifications away from the cyclone: the shift of the 1028hPa contour over southwest WI, the 1024hPa contour over eastern MI and northern FL, the 101hPa contour over eastern NC and VA, and the strength of ridging over the water off the southeast coast. Resolution is probably not the primary cause of these differences.

• Initial condition differences due to the digital filter may have resulted in these forecast differences at 12 hours

• Adjustment due to initial imbalances may have caused the model to move toward a different large-scale state. Not only is gravity-wave noise generated, but also the large-scale balanced fields can be modified. Experiments in other cases without the digital filter found even bigger large-scale adjustments
  

How imbalance generated by physics forcing affects the model forecast

The model responds to intense sudden changes by sending out gravity waves and beginning the adjustment process to bring those changes into "balance." A dramatic example caused by intense latent heating from the grid-scale precipitation parameterization is shown here.

This 12-hour forecast plot from a hydrostatic 10-km Eta threats run shows 500hPa vertical velocity in microbars per second, negative for upward motion. The white area in the center of the ring pattern is rising at more than 40 microbars/second. Instead of the grid-scale precipitation scheme steadily releasing latent heat over the eight minutes between the times it updated, this older version of the model was releasing all that latent heating in 24-second bursts. The model responded to the sudden changes with gravity waves. Cloud models (such as with grid spacing of 1 km or smaller) respond to the initiation and collapse of vigorous updrafts with a display of outward-propagating gravity waves, and satellite pictures of convection sometimes reveal waves in nature as well, though in both a cloud model and nature, the wavelength and area affected are smaller than in the example shown here.

This graphic shows the corresponding 850hPa heights and absolute vorticity. Only vorticity greater than 16 x 10^-5 s^-1 is shaded, with red exceeding 50 x 10^-5 s^-1. The gravity waves show up primarily in the height and vertical motion/divergence fields, not so much in vorticity.

This figure shows the same 500hPa plot but from a parallel run using the new grid-scale precipitation scheme implemented in November 2001. The area of strong upward motion in central Texas is similar but the latent heating is applied more smoothly in time. There is still a hint of gravity wave rings drawn in black.

This figure shows the same 850hPa plot from a parallel run using the new grid-scale precipitation scheme.

As a forecaster, you need to note several things:

• The gravity wave noise masks the mean signal. This is not only true of the model: raw (not time-averaged) wind profiler observations tend to be dominated by short-period vertical motion oscillations that mask the weaker large-scale signal

• The gravity waves may modulate other features in the model, causing them to appear to line up with the waves

• The most important forecast impact is what happens to the feature once its forcing ends. In this case, what will happen over 12 hours, 24 hours, even 72 hours, to the vorticity and height perturbations associated with this precipitation bull's-eye over south-central Texas once the precipitation weakens? The answer will be the same for both the model run with the pronounced waves and the cleaner run with the newer microphysics. It all depends on the adjustment process — whether the feature of interest (the main feature, not the waves) is large or small compared with 2L_R

How model resolution affects the forecast impact of spurious model features

Model resolution affects the scale of the disturbance and the adjustment process.

• Features like convective systems and sea breezes may be represented but are too broad in a coarse-resolution model run

  • Features tend to be more "balanced" and dynamically "large," with longer lifetimes and more influence on the forecast in a coarse resolution model than in real life

• Very high resolution models (say, 1 or 2 km) have finer scale structure and adjust with smaller scale gravity waves, which propagate more of their energy vertically

  • The adjustment process does not spread the influence of small-scale features as far

  • Spurious model features tend to be short-lived with relatively little impact on the overall forecast evolution
    

This is why an episode of spurious grid-scale convection can strongly influence synoptic features in the AVN model forecast while having far shorter and more locally confined impact in the Eta model forecast.

7.0 Application to Data Assimilation

When the model run starts with a new analysis, imbalances in the initial conditions cause gravity wave noise early in the forecast and an adjustment of the large-scale fields. This is described in section 6.0. It is why most analyses apply some sort of balance constraint and why an initialization procedure may be run following the analysis.

At this point you may want to review the clickable diagram on Page 4, in the "DA Process" section of in the COMET module, "Understanding Data Assimilation," at http://meted.ucar.edu/nwp/pcu1/ic6. This provides a refresher on the terminology and process of converting observations to observation increments (difference between observations and the first guess forecast) and analyzing those into analysis increments (differences between the analysis and the first guess forecast).

For models with an assimilation cycle (such as the operational Eta in 2002, but not the nested higher-resolution runs), the forecast impact of new data is the model's response to the analysis increments (changes from the first guess). Thus, what matters is whether the analysis increments are primarily in the wind field or the mass field and whether they are large or small in scale compared to our favorite length scale, 2L_R. This means:

• In the tropics, the forecast can be most improved with wind observations

• In middle latitudes, the large-scale forecast winds will be determined largely by the initial mass field, making temperature observations most important

• Moisture observations don't create much dynamic adjustment by themselves but are linked to both mass and wind changes through the model integration during the assimilation period, during which period the model evolution is affected by changes in radiation and latent heating in model precipitation
  

• How the data are put into the model makes a huge difference

  • Forcing the analysis to tightly fit observations, especially of pressures/heights/temperatures, in a small region around the observation will not have much lasting impact on the forecast and may produce adjustment toward an incorrect large-scale state, resulting in degradation after some time into the forecast

  • Mesoscale data assimilation is a major challenge, and getting consistent forecast benefit beyond a few hours from assimilating mesoscale observations is extremely difficult
    

Forecast impacts of analysis correlation lengths

The scale of analysis increments, and thus their impact on the subsequent model forecast, depends heavily on some details of the analysis method rather than just depending on the scale of observed features or the observation spacing.

Statistical analysis systems, including 3D-VAR, assume that there is some relationship between corrections required to the first guess (also called "background") some distance from an observation and the correction required at that observation location. This relationship, properly expressed as forecast error covariances (also called "background error covariances"), is characterized with a correlation length, which is a distance over which the relationship extends. To review how 3D-VAR works, refer to the COMET module, "Understanding Data Assimilation," mentioned previously, especially the graphic at http://meted.ucar.edu/nwp/pcu1/ic6/6_5d.htm.

Correlating the correction to the first guess over some distance produces smoothing of the analysis increments. Both vertical and horizontal relationships are used, so there is effectively some smoothing in both the vertical and horizontal. Typically, correlation lengths will be longer for coarser resolution models, meaning small-scale features poorly depicted in the first guess field will show up at larger scales in the analysis than in reality.

The horizontal correlation length affects the horizontal scale at which information from observations enters the analysis. If this length scale is large, only synoptic-scale features will be changed by observations, and the mass observations will be most critical to the forecast. If the correlation length is short, then wind observations become more important because the analysis increments will involve more small-scale structure.

The vertical correlation length affects the depth through which observations influence the analysis, and thus the Rossby radius of features in the analysis increments. If the vertical correlation length is short, then the Rossby radius will be smaller, making mass observations more useful.

Remember, the analysis comes from adding the analysis increments to the first guess. Thus, detailed structure in the first guess forecast will get passed along to the analysis if the influence of observations is smoothed to coarser vertical and horizontal scales.

Assimilation of high-resolution observations in high-resolution models

This subject is explored through some challenging questions on a special advanced subtopics page. The content on that page is not tested in the exercises at the end of this module or in the LMS exam. You are encouraged to let your curiosity get the best of you and try it — even if you don't select the correct answers, you will learn from the discussion. Before proceeding, make sure you understand the concept of correlation length explained above, then continue to the special advanced subtopics page.

Consider two models of different resolution using the exact same analysis system:

• Each model has its own assimilation cycle
• Each model has its own analysis
• The correlation lengths and other aspects of the analysis method are identical for the analyses for both models
• The correlation lengths and other aspects of the analysis method are more appropriate for the scale of the coarser-resolution model

Please answer True or False for each of the following statements. When you are online after each question you receive an individual explanation. This advanced subtopic lives at
http://meted.ucar.edu/nwp/pcu1/D_adjust/modassim2.htm

(True or False) a) High-resolution observations will contribute to more detail in the analysis in the higher-resolution model.
(True or False) b) If both models produce the same number of spurious features in their forecasts, the higher resolution model will have more spurious features in its analysis.
(True or False) c) The sea-breeze convergence boundary will have better placement and strength in the initial analysis of the higher resolution model.
(True or False) d) If the higher resolution model produces better fits to the observations during the first few hours of the forecast, it will have less run-to-run forecast variability.
(True or False) e) If the higher resolution model gets a new analysis system with shorter correlation lengths and it makes good use of high-density observations to create analyses with accurate small-scale details, then this will all result in large improvements in 36-hour forecasts.

Appendix A - Complications to Gravity Wave Speed

8.0 Exercises

You can take these questions online and get expert answers for each one. The exercises begin at http://meted.ucar.edu/nwp/pcu1/d_adjust/8_1.htm.

8.1 Exercise 1

To determine if an observed or model-predicted feature is "large" or "small" compared to , you need to know its: (Select all choices that apply.)

		a) altitude
		b) average vorticity
		c) average lapse rate
		d) intensity
		e) vertical depth
		f) location
		g) latitude
		h) size

8.2 Exercise 2

Choose the appropriate "completing word," either "Temperature or Wind."

	A feature determined to be "Large" (compared to ) will retain much of its (Temperature or Wind) field while the
	(Temperature or Wind) will adjust to come into balance with it.

8.3 Exercise 3

Balanced temperature and wind fields achieved after dynamic adjustment in both reality and numerical models create an atmospheric state that is... (Click your choice to complete the statement.)

a) geostrophic, because the mass and wind fields always are adjusting toward geostrophy. b) ageostrophic, to be consistent with latent heating, jet streak and frontal circulations, curved flow and friction. c) ageostrophic, because the mass and wind fields are independent. d) ageostrophic, because it includes gravity waves.

8.4 Exercise 4

Select True or False for each of the "completing statements" below. Thinking of features of increasing size from small to large scale, the Rossby radius of deformation (Lr)...

(True or False) a) is the scale at which rotation becomes less important than buoyancy.
(True or False) b) is the size beyond which features tend to be rotational in character and long-lived.
(True or False) c) is the scale at which rotation becomes as important as buoyancy.
(True or False) d) is the size where features larger than this are usually dispersed by gravity waves.
(True or False) e) increases proportionally with the depth [vertical size] of the disturbance.
(True or False) f) increases with increasing stability.

8.5 Exercise 5

There is a continuous range of adjustment behaviors between the "small" and "large" extremes for intermediate-size features. How does improving model resolution affect the forecast impact of a spurious forecast feature 7 times larger than the grid spacing? Write your answer in the space below.