A Warming Arctic Would Not Cause Increased Severe Weather or Temperature Extremes

Guest essay by Chuck Wiese, Meteorologist, Weatherwise, Inc.

This essay is a critique Francis and Vavrus (2012), hereinafter FV (2012), by atmospheric scientists Jennifer Francis from Rutgers University and Steve Vavrus of the University of Wisconsin. Their paper can be downloaded here:


and an updated version here:


FV (2012) claims a measured decrease in the zonal or west to east wind component due to “arctic amplification” (AA) would increase jet stream meandering, increase the amplitude or “waviness” of the flow, and increase persistent long wave blocking patterns around the northern hemisphere. This, in turn, would increase severe weather, droughts, floods and temperature extremes.

To quote the authors directly:

“Two effects are identified that each contribute to a slower eastward progression of Rossby waves in the upper-level flow: 1) weakened zonal winds, and 2) increased wave amplitude. These effects are particularly evident in autumn and winter consistent with sea-ice loss, but are also apparent in summer, possibly related to earlier snow melt on high-latitude land. Slower progression of upper-level waves would cause associated weather patterns in mid-latitudes to be more persistent, which may lead to an increased probability of extreme weather events that result from prolonged conditions, such as drought, flooding, cold spells, and heat waves.”

To quote the authors again, the effects described above are the result of arctic amplification, a term defined by the authors as:

“Arctic amplification (AA) – the observed enhanced warming in high northern latitudes relative to the northern hemisphere”

This definition seems to fit the claims made by NASA GISS and NOAA that temperature measurements of the arctic are warming at a much greater rate than anywhere else in the northern hemisphere.

To examine these claims by the authors, I will use an application of dynamic meteorology from atmospheric science and introduce the physics of Rossby waves, invoked by the authors as applicable to validating their claims as well as a few of the governing laws of motion that describe the behavior of these waves and how they would interact with a warming arctic.

The physics of wave motion can become a very math intensive discussion. For the sake of article simplification I will provide the steps of deriving these governing equations which used to be used in synoptic meteorology as a tool by forecast meteorologists in the appendix page of this article with use of the resulting equations to estimate how a particular wave pattern in the atmosphere will behave in a generalized sense.

This was before modern computing power enabled meteorologists to expand on these same ideas and obtain through that computing power a very complex and comprehensive set of equations that can deal with each hemispheric weather system and its associated Rossby wave individually and affect the answers of all in much greater specificity. BUT, this does not affect the purpose or conclusion derived in writing this article whatsoever.

Two terms come into the discussion which are important to define. They are the amplitude and length of a wave, called amplitude and wavelength respectively. In the diagrams below, the amplitude of a wave is the amount of northward and southward stretching it can assume in time, which would be along the clip_image018 axis in Cartesian coordinates, paralleling the north and south component of the wind, clip_image020. It is a measure of the clip_image018[1] or north-south distance between a ridge and trough axis.

A high amplitude wave or flow means a greater clip_image018[2]clip_image021distance between the ridge and trough peaks. These are called long waves in the westerlies and are considered full latitude waves often starting as low as 30 degrees north latitude and extending as high as 80 degrees north latitude. The wavelength is defined as the lateral spacing between the waves over 2clip_image023 radians since the waves are of a trigonometric form and are measured along the clip_image025 axis or west-to-east direction in Cartesian coordinates, paralleling the west to east wind component, clip_image027 , referenced by FV (2012) as the “zonal wind flow”.



FV (2012) claims Rossby wave physics shows arctic amplification (AA) and weakened horizontal temperature gradient will decrease the westerly or zonal wind component in the arctic. This decrease causes atmospheric waves to increase in number, amplify and stall over particular regions of the earth. This stalling increases severe weather, floods, and low temperature extremes under the troughs, and droughts with high temperature extremes under the ridges.

To examine FV (2012)’s claim, we need to define Rossby waves and look at how AA might change how Rossby waves behave. In the Appendix, we derive and explain the Rossby equation. Here, we will use the Rossby equation to demonstrate the first flawed assumption in FV (2012).

In Appendix 3, Eq (11) we derive,

clip_image030                                                   (11)


clip_image002 = the zonal and hemispheric speed of the west winds

clip_image004 = the speed of the individual waves traveling within the flow

clip_image006 =  the Rossby parameter

clip_image008 = the length between the waves spanning 2clip_image023[1] radians

The terms that arrive from the general solution for clip_image032 in the appendix.

This is the Rossby equation, derived in 1939 by Carl Rossby, who became a famous scientist for his work in atmospheric dynamics. It is immediately apparent that based upon the computation of the mean zonal speed of the westerlies across the hemispheres, that a good idea can be surmised as to how many planetary waves we could expect to set up based upon the use of this equation.

For easterly moving waves which have a positive value of clip_image034,


because clip_image038  and clip_image040.

For westerly moving waves which have a negative value of clip_image034[1],


because clip_image038[1]  and clip_image044, and the length of the waves would be relatively long instead of relatively short.

In other words, the higher the wave speed compared to the zonal current, the shorter the wave lengths.

As the waves slow in progression, we see that values of  clip_image046 approach clip_image048 and we get much longer wave lengths for a given value of clip_image048[1], hence less planetary waves around the hemispheres. The longest wave lengths are apparent when the waves actually retrogress in the atmosphere, in other words, move from east to west. In such a manner, we then have a negative number for clip_image034[2]clip_image049and it is obvious the value of clip_image046[1] becomes greater than clip_image048[2] for the maximum wave length permissible.

In terms of the speed of the waves, the Rossby equation demonstrates that FV (2012) is incorrect, because as the waves slow with respect to the zonal winds, the wave lengths increase, which is the opposite of what they claim because that result decreases the waviness of the flow around the hemispheres. But what the authors also argue to claim more waviness to the jet stream (counter intuitive to the wave speed) is that the speed of the current clip_image048[3] or jet stream speed is declining due to arctic amplification weakening the horizontal gradient of temperature across the latitude lines.

To examine this, we will further simplify the Rossby equation by assuming the persistent scenario claimed by FV (2012) of stagnating weather patterns giving weather extremes results from the waves becoming stationary or standing in the flow which they occasionally do. We then set  clip_image051 for wave speed and solve the Rossby equation for clip_image053 . The result is we get an arbitrary maximum wave length clip_image055 that depends on the zonal jet stream speed and latitude from the Rossby parameter


While it is apparent from this equation that the length between the waves can decrease from a declining zonal wind speed clip_image048[4] as claimed (a more wavy flow) it is also apparent that based upon the measured decline of the westerly winds in the data estimated by the authors (about a 14% decline) that this does not increase the number of waves with any weather significance around the hemisphere if one considers that impact over the range of speeds we find in the real atmosphere. The most common pressure level that is closest to a non divergent level required for use in the Rossby equation is near 600 millibars or just under 18,000 ft. The atmospheric range there is between clip_image058.

Using this equation, I prepared a table below (Figure 3) which shows us the number of permissible standing waves based upon the speed of the zonal current at latitude 45 degrees North and 60 degrees North over the normal range of speed found and then with a 14% reduction. As we can see, it makes little difference and the number of waves or “waviness” in the flow remains nearly the same for most of the westerly wind speeds. So again, the authors are incorrect according to the Rossby physics.


Critics of the use of this equation will try and argue that it is over simplified in explaining the behavior of a chaotic system such as the earth’s atmosphere and that wavelengths are much more unstable and non uniform in nature, unlike the limitation the mathematics places on the use of this equation. While part of that is true, it is also true that we do not use this equation alone to describe wave behavior in the atmosphere in an absolute sense.

Phasing equations and other physical equations determining the more complex motions of the atmosphere not related entirely to Rossby physics are used in weather models to incorporate the more complete set of governing behavior. BUT it is absolutely true that in a climatic sense such as what is claimed in FV (2012), this equation derived by Rossby gives us an accurate portrayal of how the waves around the hemispheres would behave and change in a general sense if their claims were true, for this equation is an important building block that sets the foundation for all of the other wave behavior and we see clearly in this part of the article that the waves would not behave as the authors claim in their peer reviewed paper.

The lateral spacing of the earth’s planetary waves or wavelengths is only half of the problem at hand to check on the validity of the claims in FV (2012).

The second and just as critical component is examining how the amplitude of these waves would change if the westerly winds across the northern hemisphere began to slow because of AA or Arctic Amplification warming as the authors claim is happening, which according to their claims would cause the amplitude of the waves to increase.

Platzman (1947) derived an expression as an aid to solving for wave amplitude by deriving a trigonometric form that allows for the conversion of the earth’s geometry in lateral spacing along latitude circles to project onto a mapping in Lambert conformal or stereographic coordinates. That can then be plugged into the Rossby physics as we show in the Appendix to derive an expression in spherical coordinates that can be solved for the amplitude of the Rossby waves.

In Appendix 2, we derive


Immediately, we can see a remarkable similarity of this expression versus the Rossby equation that describes the length of the waves. If we set aside the trigonometric terms for a moment, we see that in a standing wave format described above by clip_image055[1], the only difference is we are not summing clip_image023[2] radians but just twice the quantity of the square root of clip_image062 multiplied by the trigonometric terms that size the amplitude.

Then clip_image064 is substituted for clip_image048[5] meaning that instead of a prior prescribed and mean or steady westerly wind belt, we now have a total and point specific wind velocity streamline vector whose wave amplitude is dependent not only on the magnitude of clip_image064[1] but on the DIRECTION of the wind vector that is described by the variables clip_image066 and clip_image068. The subscript “1” refers to the wind direction at what we call the inflection point latitude of the waves, or when the wind streamlines make a clear break in direction from a trough to ridge axis.

Likewise, clip_image070clip_image021[1]and clip_image072 are the maximum amplitude latitude and inflection point latitudes respectively of the waves which are typically found to be roughly one half of the amplitude of the waves from trough to ridge axis, as a rule of thumb. Now the variable clip_image074 or amplitude no longer has to be prescribed as a constant as it was in the case of the equation that described Rossby wavelengths, but varies with wind speed, direction and latitude, all prescribed in the above equation.

If we compare the physical meaning of what defines Rossby wavelength and speed in these equations, the corollary is clear. As the wind speed increases along the given waves, so must the amplitude and wavelength. As speed decreases, so must the amplitude of the waves and their respective wavelengths, with the maximum amplitude of the waves being realized when the inflectional direction has a backed wind direction to south or even east in the case of closed off low pressure streamlines. THIS IS CLEARLY IN CONTRADICTION AND OPPOSITE OF WHAT IS CLAIMED BY FV (2012).

To illustrate this point, let us take a case of demonstrating what happens to the wind direction and speed of the jet stream when it is exposed to increasing gradients of temperature, or the opposite effect of what FV (2012) claims is happening in their paper.

Have you ever wandered outside, looked up into the sky and felt the wind blowing against your back or face and then when looking up at passing clouds that they are moving from a totally different direction from the wind that blows against your face or  back?

This happens frequently in the real world and is a visual example of what happens to the winds going upward in the vertical when the wind and its associated pressure surfaces no longer parallel the isotherms. When this happens, the atmosphere leaves the state of a more energetic stability and increases and liberates potential energy as the wind begins to blow more cross isothermal or at increasingly normal angles to the isotherms.

When this happens, the wind is now moving respective warm and cold air masses to different latitudes and longitudes and with the help of Rossby wave behavior, also allows the temperature difference over a fixed amount of space (the temperature gradient) to begin to increase. That process creates weather frontal systems that begin to generate lift and start the process of creating a low pressure system or storm.

In the illustrations below, we start with a developing low pressure system so that along the stacking lines of temperature gradient we have a cold front defined by those isotherms and the gradient of temperature is taken as 2 deg C across one hundred miles of latitude. The surface wind is completely normal to or at a 90 degree angle to the isotherms as the surface wind barb indicates 30 mph of wind from the west. The isotherms are oriented north to south for a west to east temperature gradient.

As the low pressure circulation begins development, the isotherms rotate 90 degrees from east to west to north and south as depicted with the wind remaining normal to the isotherms, thus pushing them eastward.

So what does this process do to the vertical wind profile?

We find that as storms develop, the vertical wind profile will shear with height, either backing from the surface wind direction (counterclockwise) or veer clockwise which depends on whether a colder or warmer air mass is headed towards you, respectively. But the speed of the wind also changes or shears with height, increasing as you go upward, and that is what we are interested in determining as the temperature gradients increase or decrease along such a system so that we can plug the results into the Rossby amplitude expression and see how the increasing or decreasing wind speed changes the wave amplitude.

In the examples below, the isotherms are oriented north to south so we want the portion of the thermal wind equation that represents how the thermal wind which parallels the isotherms will back the surface wind and increase it with height if we maintain the temperature gradient of 2 deg C through 500 millibars or about 18,000 ft of pressure altitude. The expression we need from Hess (1959) is


Where the derivatives of clip_image020[1] and clip_image077 are taken as finite increments as clip_image079 is then the change in the northward component of the wind for a fixed increment of geopotential height, clip_image081clip_image082.

The letter clip_image084 is the earth’s gravity, clip_image086 the Coriolis parameter already defined, clip_image088clip_image089is the mean temperature of the layer of clip_image081[1] we are considering, which is 18,000 ft deep and the partial derivative of  clip_image088[1] with respect to clip_image025[1]clip_image090or the gradient of temperature on the west to east axis the isotherms are plotted on. This is also a finite quantity since it now has no other dependent variables.

The mean temperature of this layer is from a US standard atmosphere and therefore has a value of 270.25 deg K. Plugging in the relative numbers we have


The west wind at the surface is given as 30 mph so we convert to the like unit of meters per second and get clip_image093. As stated, the thermal wind is a shear vector whose top is added to the bottom value to get the resulting wind at the top so


Then the resulting wind vector at 18,000 ft has backed clip_image096from west to


or to the southwest at clip_image100.

The magnitude of this wind vector is the square root of the sum of the squares of the west and south wind components respectively, therefore


So now we have a cold front and low pressure system whose temperature gradient through 18,000 ft of geopotential height maintains 2 deg C of temperature gradient across the front over 100 statute miles and results in the west surface wind direction and speed of 30 mph backing at 18,000 ft to a direction of clip_image100[1]true at clip_image103.

With this result, notice that we now have the wind vector clip_image064[2], so that we can go back to the Rossby amplitude expression to check on what amplitude this developing storm would likely generate.

The variables are the ridge axis wind direction and latitude, clip_image066[1] clip_image104and clip_image070[1] and clip_image068[1] and clip_image072[1] (defining the wave amplitude from the inflection point) respectively for the cosine and secant.

In Fig 4 below, we must have a west wind direction, clip_image106 , for the ridge axis and the ridge axis latitude is clip_image108degrees north or clip_image110which we determine from the following steps.

At the inflection latitude, we have the variables clip_image072[2]clip_image111and clip_image068[2] which are the inflectional latitude and wind direction. Taking those from the above for cosines we have clip_image113and clip_image115.

The wind vector magnitude was calculated to be clip_image103[1] or clip_image117 . The clip_image119for the Rossby parameter is clip_image121for 50 degrees north latitude. With the inflectional wind direction, speed and latitude, you then pick an arbitrary “first estimate” latitude for the maximum wave amplitude defined at the ridge axis with a west wind for clip_image070[2]clip_image021[2].

We need to match the prediction of this maximum amplitude with the true latitudinal distance fromclip_image072[3] to clip_image070[3] , keeping in mind that what the equation is doing is integrating the effect of Coriolis turning across the latitude lines from the Rossby parameter contained within.

Therefore, if our first estimate is too small a latitude change or amplitude, the equation will predict too large an amplitude compared to the true distance between latitudes selected. On the other hand, if our first estimate exceeds the true wave amplitude, the equation will predict an amplitude too small compared to the true distance between the estimated maximum amplitude latitude and the inflection latitude. The inflection latitude was arbitrarily chosen.

After narrowing the prediction estimates to match near the true latitudinal distance, we are able to correctly calculate the maximum Rossby amplitude from the inflection point. This gives


The maximum amplitude of this wave is 787 statute miles north of the inflection point to ridge axis latitude as depicted in Fig 4 below.

A verification of this answer is confirmed by cross checking it with CAVT trajectories, calculated from inflectional data by Hess & Fomenko (1955).

Note that we are not attempting nor does this equation predict the meridional displacement of the wave. For that can be easily referenced from the tables and we find from this that the amplitude peak is displaced 36 degrees of longitude east of 130 degrees west or at 94 degrees west longitude.

According to FV (2012), as the zonal winds decrease, we are supposed to see slowing progression of the waves with the amplitudes of the waves INCREASING as the zonal winds decrease from AA or Arctic Amplification warming BECAUSE OF THE DECREASING GRADIENT OF TEMPERATURE.

So in the first example given in Fig 4, we have an amplitude from the inflection point of 787 statute miles and the upper level wind speed at 18,000 ft of V = 58 mph. This results from a temperature gradient of 2 deg C per 100 miles of west to east distance as depicted and maintaining that gradient through 18,000 ft of geopotential height.

So if we increase the gradient of temperature in Fig 4 by 2.5 times 2 deg C to 5 deg C per 100 miles of west to east distance along the same front with the same west wind direction increased to 40 mph at the inflection latitude, we can repeat the calculations and check the answer for amplitude.

The result is that because the gradient of temperature is increased by a factor of 2.5 the wind vector clip_image064[3] at 18,000 ft now increases to 130 mph and backs yet further to 72 degrees from west or 198 degrees true at 130 mph as depicted in Fig 5. This result if FV (2012) is correct should give us a lower amplitude wave value compared to Fig 4.

Repeating the calculating steps from Fig 4 for Fig 5 gives


The maximum amplitude of this wave is 1,263 statute miles from the inflection point latitude. This is 60.5% greater than with the weaker temperature gradient depicted in figure 4, the exact opposite of what is claimed will happen in FV (2012) from decreasing winds due to decreasing temperature gradient from AA or Arctic Amplification warming. FV (2012) is clearly incorrect.



To further solidify these ideas, I invite the reader to examine figure 6 below. This diagram is well appreciated by meteorologists and describes the changing behavior of the atmospheres circulation as it goes through the accumulation and conversion phase of storing potential energy across the latitude lines caused by warming surplus energy from the sun at lower tropical latitudes all year that are pitted against the changing loss of this energy at the poles due to the axis of rotation of the earth tilted at 23.5 degrees with respect to the sun.

The diagram has four phases from A to D. The first phase in (A) is called the high index phase. This occurs when the latitudinal gradients of temperature are low. This phase is exactly what FV (2012) claims the earth is headed towards due to rapid warming of the Arctic caused by their claim of “Arctic Amplification”.

It is called zonal flow in meteorological terminology. Notice the waves are flat and have a low amplitude as the equations tell us we would get. The speed of the jet stream is also slower in this state due to the weaker gradients of temperature across the latitude lines. Over time, and especially towards the season of winter, a much more rapid loss of energy to space by radiational cooling occurs near the polar region while the tropics continue to accumulate excess heat energy from the higher sun angles. As this temperature imbalance increases, so will the temperature gradients across the latitudes and speed of the jet stream.

Eventually, the travelling, shorter Rossby waves in this accelerating flow have a sufficient amplitude because of reaching a critical speed for baroclinicity begin to transport heat energy poleward from the tropics. This begins the process of cyclogenesis or storm development in the troughs and we begin the change to phases (B) and (C) that are called low index flow.

Jet stream speeds in this phase continue to INCREASE, not decrease, thus increasing the amplitude of the waves as the equations tell us. This continues until we reach phase (D) where individual closed circulations of high and low pressure occur. At this point, the maximum exchange of heat energy towards the poles and cold air from the poles towards the equator occurs and the storms and high pressure cells reach their maximum intensities and amplitudes.

Notice in this phase, just as the amplitude equation calculates, the maximum amplitudes not only occur with the higher wind speeds, but additionally from backing wind directions that exceed 90 degrees of deflection, or directions that begin to obtain an easterly component, often indicating the waves are retrogressing westward.

This process then begins the process of relaxing the latitudinal temperature gradients and from there, the Rossby wave physics begins deforming and filling the low pressure cells and the amplitude of the flow reverses back to high index and low amplitude with an initially stronger westerly jet stream due to its displacement to a more southerly latitude. As the temperature gradients continue to weaken, so does the jet stream and it once again begins migration to higher latitudes and starts repeating the process again beginning with phase (A).




FV (2012) cited in the introduction of this article is fatally flawed, incorrect and should be withdrawn by the authors. As shown here, there is no theoretical basis in which to ground FV (2012). Using the proper Rossby wave physics as illustrated here, these atmospheric waves (or commonly called planetary atmospheric waves that generate low and high pressure systems that create our weather, severe and otherwise) behave in the opposite fashion as claimed in FV (2012).

A warming Arctic that is supposed to be weakening the westerly wind belt across the northern hemisphere would create an entirely different effect on the earth’s weather as FV (2012) claims. If FV (2012) claims were true, the physics governing these waves would require them to flatten in amplitude and migrate to a higher latitude, causing a much weakened effect on the Northern Hemisphere’s weather patterns.

If FV (2012) claims were true, precipitation systems would weaken and migrate northward with the migrating jet stream. Storms, severe and otherwise would become far less common than today and would be replaced with problematic drought and much higher surface absolute and relative humidities. This increased low level moisture would lead to sporadic showers and thunderstorms in an ever expanding maritime tropical airmass environment, but not enough precipitation to forestall severe droughts.

By severe droughts, I don’t mean regional droughts such as those experienced recently in California. But rather, droughts that would expand into a worldwide regime. Present-day droughts are nothing more than cyclical changes in the earth’s climate system that have very definitive and repetitive cycles.

What is particularly disturbing about FV (2012) is not only is it incorrect and flawed, but it passed peer review. Now, after publication, FV (2012) has been lapped up by media, touted and referenced in their severe weather stories that report on hurricanes, tornadoes, severe thunderstorms, heat, cold, drought and any other weather calamity as “proof” their paper is correct. Nothing could be further from the truth.

The reader needs to understand that anytime we experience severe weather, it is proof that adequate COLD in the high latitudes and Arctic has been generated by the normal radiational cooling process by the earth that creates the adequate potential energy across the latitude lines to cause amplification of the jet stream waves and speeds that pushes this colder air southward to warmer latitudes that then creates the necessary temperature gradients to liberate that energy, creating storms as well as high pressure systems.

If the occurrence of severe weather is increasing worldwide, it is not a sign of a warming earth. It is the opposite of what climate hysteria claims, and an indication of a cooling, not warming earth.

The continued misuse, abuse and general trashing of important principles founded with atmospheric science remains as deplorable as ever by the groups promoting global warming from human CO2 emissions or by these same groups promoting climate hysteria by re-labeling this term “climate change”.

Now that the flawed FV (2012) passed peer review, it allows media to blame any severe weather on “climate change.” FV (2012) allows media to claim a wavier jet stream dips and meanders because the Arctic is supposedly getting warmer. All this is sheer nonsense and all demonstrably wrong.

I believe this flawed FV (2012) also shows how the quality of the scientific peer review process has been lowered in “climate science”.



Large atmospheric waves as analyzed and seen on synoptic weather maps behave according to their derivation from the atmospheric vorticity theorem. Vorticity is another term used in atmospheric dynamics that describes spin motion characteristics in a stream flow of air such as the jet stream. The spinning behavior of the air in such a flow is generated by the spherical geometry of the earth and the earth’s rotation by itself as well as speed shearing along these rivers of air that surround the earth at higher altitudes.

This introduces terms such as the Coriolis force and the Rossby parameter, each assuring because of the earth’s rotation that the absolute vorticity of the earth be conserved across the lines of latitude from equator to pole, or defined as



clip_image010 = the Coriolis parameter, defined as




clip_image012 = the angular speed of the earth = clip_image131xclip_image133

clip_image014 = the selected earth latitude

clip_image016 = the relative vorticity about the vertical axis at any point on the earth and is defined as


where clip_image027[1] and clip_image020[2] are the respective west to east and south to north wind components.

So the time derivative of the sum of clip_image086[1]clip_image136and clip_image138 being equal to zero means that if the relative vorticity about the vertical increases at a point, the Coriolis parameter must decrease an equal amount, which as we see is latitude dependent, so the rate of change of absolute vorticity at any point across the latitude lines is conserved and always zero but changes relative to the latitude, which defines the relative vorticity.

In the derivation of the Rossby wave equation, it should be noted that we need to use the rate of change of the Coriolis parameter, clip_image086[2] , so we take the derivative of clip_image086[3]with respect to clip_image140, then in Cartesian coordinates,



a = the radius of the earth.

Note that clip_image018[3] represents the north-south axis which represents the changing lines of latitude from equator to pole and the derivative of clip_image086[4] is then divided by a which is the mean radius of the earth.

This gives us the change in the latitudinal dependent relative vorticity about the vertical in dimensions of clip_image144 because we divided by a.

An important physical characteristic of what we have done so far is to make it clear that the relative vorticity about the vertical increases with decreasing latitude. This has a significant meaning to the development of long waves and storm systems in the westerlies in that any wave cyclone or low pressure system that propagates along such a wave is subject to a “spin up” or intensification if it moves to a lower latitude and gains relative vorticity about the vertical. Likewise, northward moving systems lose relative vorticity to Coriolis turning and are subject to spin down or weakening.


Now we turn to the construction of the Rossby wave equation used in atmospheric science to check on the validity of the claims made by FV (2012). Assumptions need to be made to this construction that simplify the mathematics considerably, because we find if it is derived from the vorticity equation it winds up being a second order, non linear partial differential equation with a product of dependent variables because of the separate clip_image090[1]and clip_image049[1]wind components and Rossby parameter.

It is simplified considerably by making some assumptions that are actually beneficial to what meteorologists are interested in knowing about the behavior of these waves as they occur in the earth’s atmospheric system. And because that is very large, we can eliminate some of the cross dependence in the equation by making the individual derivatives follow a point in the stream rather than a parcel of air directly so that the waves have a constant shape and follow a large river of air around the hemispheres that we actually see and define as the westerlies that encircle both hemispheres.

That large river of air is then taken as clip_image048[6]rather than a localized clip_image027[2] component and disturbances in the flow are perturbed by introducing a clip_image020[3]clip_image049[2]component, clip_image032[1] of velocity into the flow that is a function of the clip_image025[2] axis (west to east) and time, clip_image146. As stated above, absolute vorticity on the earth is conserved so that from Hess (1, 16.4, 16.5)


If we describe the wind components to consist of a broad westerly wind current with a north/south wave pattern of infinite lateral extent, then the dependent variable can become independent of clip_image018[4] so that we have a system of waves in which the streamlines at any latitude are parallel to any other latitude. Then




If one were to follow a point moving in the west-east direction with speed clip_image034[3], no changes will be observed in any of the variables. That is the operator


clip_image152 is an individual derivative following a point moving with speed clip_image034[4]clip_image049[3]. This is different than the derivative clip_image154 that would follow a parcel of air directly. With these assertions, the vorticity equation becomes


This is a difficult to solve nonlinear equation with a product of dependent variables. To simplify it we assume the above conditions describing clip_image027[3] as a large river of west to east moving air clip_image048[7] encompassing the entire hemisphere, with superimposed and smaller perturbations of clip_image157 and clip_image032[2] travelling within in it that are functions of clip_image025[3] and clip_image146[1]clip_image158. The nomenclature is then


Then clip_image048[8] becomes the zero order of magnitude because it is very large compared to clip_image157[1] or clip_image032[3]as we choose to make those planetary waves about 1/10 as large as clip_image048[9]. So the perturbations are then a first order of magnitude to clip_image048[10] as they are an order of 1/10 on a logarithmic scale, and the additional resulting terms that follow from this rearrangement are an order of magnitude smaller yet, or 1/100 of clip_image048[11] on a logarithmic scale and because of this are of a second order of magnitude. We then end up with the following linear second order partial differential equation with constant coefficients


With the above stated analogy, the first two terms are then a first order of magnitude to clip_image048[12] but the third is a product of the first order magnitudes making it a second order magnitude or 1/100 as large as clip_image048[13]. The result of this is that the third term is sufficiently small to ignore. We can now simplify this differential equation to


Where clip_image119[1] is the Rossby parameter we have already defined and the newer term clip_image034[5] is the speed of the waves. A general solution to this equation as we have defined the functions is trigonometric and we have a solution with clip_image163 representing the maximum wave amplitude from trough to ridge axis. Then


Substituting for clip_image032[4]




Which then reduces the differential equation to


This is the Rossby equation that was derived in 1939 by Carl Rossby that describes the frequency of atmospheric waves by their lengths that become a function of their speeds from both the general speed of the westerly jet stream surrounding the earth and the speeds of the smaller waves themselves that traverse within the larger river of air called the westerlies.

One of the disadvantages of this equation is that it does not speak directly to the amplitude of these waves even though the maximum amplitude is specified in the general solution. That must be assumed a constant in the formulations described and vanishes as such in the final solution. I want to address this part of the problem with much greater specificity, so I will introduce more dynamics for a definition of the wave amplitudes.


As the Rossby waves are introduced in the prior solutions, the amplitudes like the wavelengths themselves must be a function of Coriolis turning as defined above. Martin (2, 3) notes this in writing the expression that the ratios of wind velocity along a streamline to radius of curvature of the flow is described by


Where clip_image064[4] and clip_image170 are the wind speed and radius of curvature of the flow respectively and clip_image086[5]and clip_image172 is the Coriolis parameter defined at an initial point along clip_image170[1] and a nearby point clip_image174, respectively.


Which describes the magnitude of Coriolis turning and the prescribed latitudes clip_image070[4] and clip_image072[4] that affect it. Omega has already been defined as the angular velocity of the earth. clip_image177 is also equivalent to the Rossby parameter and product of latitudinal displacement which may be written as


Recalling and defined previously that the Rossby parameter is clip_image180. The new variables are now clip_image018[5]clip_image021[3]and clip_image182 that define the latitudinal displacement of the wave along a particular meridian which defines the amplitude along the meridian or  axis in Cartesian coordinates. The objective in quantifying the amplitude of Rossby waves is also to make the “flat earth” Cartesian coordinates to a spherical form that represents the true latitudinal distances of the earth. Platzman (1947) had achieved this by showing that if clip_image184 is the radius of the earth as earlier defined, then


where clip_image066[2] is the wind direction measured relative to a latitude circle on Lambert conformal or polar stereographic coordinates.  Combining (12) and (15) yields




Resulting in


The first order approximation is




Equation (17) is useful now because it can be further resolved in aiding the construction of a Rossby wave with the appropriate data. Namely, this equation can be further resolved to provide a definition of inflectional latitude and inflectional wind direction as defined by Martin (3) providing an inflectional streamline exists, which by definition will exist providing that


This would tell us that the cyclonic curvature radius on the left term must be larger than the effects of Coriolis turning on the right hand side. In most synoptic scale systems, we find this is almost always the case. But we are not interested in the actual construction of the complete wave, but rather, we want an accurate assessment of the wave amplitude regardless of the meridian that the ridge or trough axis positions itself on. The point of this article is to tie in wave amplitude with wind speed and wind speed to the claimed weakening latitudinal temperature gradients and cross check those computations with the claims made in FV (2012).

To do this, we can simplify (17) further by taking the arbitrary initial point as the inflection point or latitude which allows us to set the third term in brackets on the right hand side of (17) to zero. We would then be computing the wave amplitude from the inflection latitude point rather than from the full trough axis that involves . This would then be approximately one-half of a full trough to ridge amplitude. This is perfectly acceptable because the ridge axis is what these authors claim is expanding northward due to AA or Arctic Amplification warming. They claim this process is “stretching” the waves in amplitude and stalling them out leading to extreme and persistent weather events.

In reality, we would find that deepening or intensifying troughs most always carry higher jet speed strength as well that takes us to the inflection latitude to compute the wave amplitude from that point. So the trough axis often expands to a lower latitude from deepening as does the northward stretching or shrinking from wind speed along the streamlines. Following this procedure, equation (17) simplifies further and can be solved for clip_image193.

The result is


This is the equation we want to cross check the Rossby wave amplitude. As a reminder that this equation does not give us a full trough to ridge amplitude, the subscripts clip_image074[1] and clip_image195 are introduced to define the amplitude from the inflection latitude respectively, not clip_image196 as was used in the Rossby wavelength amplitude that defines the maximum amplitude selected from trough to ridge axis. Likewise, the initial points clip_image072[5]and clip_image068[3] now become the inflectional latitude and inflectional wind direction respectively and clip_image066[3]and clip_image070[5] are the wind direction at the ridge axis and ridge axis latitude respectively. At the ridge axis, the wind direction would always be from the west so that clip_image106[1]. Then any arbitrary inflectional latitude, wind direction and speed can be chosen.

In this article, I selected 50 degrees North latitude but the wind direction and speed at this point was computed by changing the temperature gradients across an arbitrary frontal boundary to increase or decrease the directions and speeds accordingly to tie together the effects of temperature gradients, wind speed, direction and Rossby wavelength and amplitude to verify or nullify the claims made in FV (2012).

The set of equations to do this are now complete. However, I advise caution when attempting to use this equation in calculating low latitude wave amplitudes. Below 40 degrees north latitude, the importance of radius of curvature begins to dominate Coriolis turning and can result in higher wave amplitude projections resulting in a greater margin of error compared to the published actual CAVT trajectories.


1. Francis, J. A., and S. J. Vavrus, 2012: Evidence linking Arctic amplification to extreme weather in mid-latitudes. Geophys. Res. Lett., 39, L06801, doi:10.1029/2012GL051000.

2. Hess, 1959: Introduction To Theoretical Meteorology. Library of Congress Card Number 59-9118, Chapters 12.8, 16.4, 16.5

3. Haltiner and Martin, 1957: Dynamical and Physical Meteorology. Library of Congress Card Number 57-8005, Chapter 20.

4. Martin, 1955: Generalization of Constant Absolute Vorticity Trajectories. Research paper number 7.

5. Hess and Fomenko, 1955: Constant Absolute Vorticity Trajectory Tables. Journal of Meteorology, Volume 1, Number 4

6. Platzman, 1947: Some Remarks on the Measurement of Curvature and Vorticity. Journal of Meteorology, Volume 4, pages 58-62.

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March 13, 2016 5:16 pm

Wow! That’s going to take a bit to digest. Nice graphics and in line with what we see from the satellites. I imagine the comments thread on this one will bring in lots of good discussion. Thank you.

Tom Halla
March 13, 2016 5:27 pm

Thanks to the California school system, I am rather illiterate in math, despite having a BA. Perhaps a collaboration in plugging the consequences of the formulae into a series of graphics, as I do not think my deficiency is unique. I think I understand the general point of the post, that the article he is criticizing is 180 off an accepted weather modeling formula, but that many formulae tend towards MEGO.

Reply to  Tom Halla
March 14, 2016 2:57 am

I don’t know what MEGO stands for but I can beat that one with MICO – Magic In, Climate Out.

Tom Halla
Reply to  Hivemind
March 14, 2016 6:42 am

MEGO–My Eyes Glaze Over. I get intimidated by formulae.

March 13, 2016 5:28 pm

.Well , that’s gonna take at least 4 Tylenol to ease the pain in my wittle bwain !!! These kind of posts should come with warning labels !! LOL

Reply to  Marcus
March 13, 2016 5:30 pm

…Caution : I.Q. over 85 required !!

Boulder Skeptic
Reply to  Marcus
March 13, 2016 10:05 pm

…Caution : I.Q. over 85 required !!

Well, that likely leaves out a number of climate scientists who appear to have no clue about the scientific method or relatively straight-forward undergraduate level statistics and math as shown repeatedly in peer reviewed paper deconstruction/falsification. I think the phrase “educated beyond his/her IQ” applies in some cases.

March 13, 2016 5:48 pm

Reblogged this on Climate Collections and commented:
Executive Summary:
FV (2012) cited in the introduction of this article is fatally flawed, incorrect and should be withdrawn by the authors. As shown here, there is no theoretical basis in which to ground FV (2012). Using the proper Rossby wave physics as illustrated here, these atmospheric waves (or commonly called planetary atmospheric waves that generate low and high pressure systems that create our weather, severe and otherwise) behave in the opposite fashion as claimed in FV (2012).

Robert of Texas
March 13, 2016 5:54 pm

Um…OK. I can’t argue the math until I spend a week or three going through this. But I do recall studies that found that in the Atlantic, storminess was correlated with Colder temperatures. To make this claim, scientists had looked at layers of sand and determined the number and approximate severity of hurricanes over the past few hundred years. They then matched this up to estimated temperatures for that period. There was a pretty strong correlation showing as the temperatures went up, storms diminished in intensity and frequency. This would be real evidence contradicting the proposed mathematical model. Or at least, one could say the hurricanes making landfall decreased in intensity and frequency.
Also, it seems intuitive to me that as you decrease the temperature gradient storms should become less severe. A warming arctic should decrease the temperature gradient.
The patterns of the upper atmosphere might or might not follow the predicted paths, but if they were to also change height, their effect might be lessened.
As for increases in drought, again I find this hard to swallow without better proof. The American West was wetter when the Earth was warmer. Colder temperatures seem to line up with increases in drought – not the other way around (talking about over hundreds of years, not tiny 10-year droughts). I have always wondered if this was not do to the Hadley Cycles lengthening or shortening, or possibly just becoming weaker.
I get what the author is trying to suggest, but so much evidence lies in front of their hypothesis that its going to take a lot to convince me. But I’ll take the time to go through the math the best I can, just in case I have a Eureka moment.

Robert of Texas
Reply to  Robert of Texas
March 13, 2016 6:04 pm

Ah! I started over reading the article and I am agreeing with the authors – I am disagreeing with the hypothesis in the original paper. My bad, I scanned too quickly the first time and thought this was arguing for increased severity and frequency of events.
The critique is exactly in line with my thinking (although its far more advanced). I based my opinion on past data on how climate behaves – their critique seems to line up nicely with this.

Reply to  Robert of Texas
March 15, 2016 9:06 am

You’ve applied the 2nd law of thermodynamics here, which those preaching the “dangers of climate change” continually ignore. I think I’ve read somewhere (can’t remember where) that climate scientists actually reject the 2nd law as applicable when it comes to a warming planet, arctic amplification, and severe weather intensity / frequency.
How can someone stand there with a straight face and tell us that both 1) the arctic is going to increase in temperature faster than the rest of the planet and 2) severe weather in the northern hemisphere is going to increase in frequency and intensity when we know that severe weather is dependent on temperature gradient? Both statements cannot be used together in discussions related to climate change, because 2nd law.
At any rate, your observation blows it away anyway. As we’ve enjoyed in the most recent “record warm” years, severe weather, at least in the US, has been below what we consider normal. For that we can THANK the warming planet and its associated arctic amplification.
We should fear a reversal of the climate trends from the last 30 years, not a continuation.

March 13, 2016 6:14 pm

I just skipped to the money quote at the end!

Sweet Old Bob
March 13, 2016 6:15 pm

IMHO this post would make an excellent video presentation .If practical , please consider doing this . (Or someone else doing it for you .)

March 13, 2016 6:19 pm

Many thanks for a clear and simple explanation of the physics of Rossby waves, better than found in some textbooks.
I had a look at H. H. Lamb’s Climate History and the Modern World. (Routledge, 2nd ed.) Lamb did not mention Rossby waves by name, but referred to them as “lobes” of the polar vortex.
Lamb referred to long-wavelength-high-amplitude Rossby waves as “blocking” events and linked them to incursions of Arctic cold air into lower latitudes. (“blocking” is the only related term in the index, pp. 36, 55, 390)
So far as I can gather, Lamb believed that the increased frequency of these blocking events was associated with the cooling observed during the Little Ice Age.

March 13, 2016 6:20 pm

The recent paper by Williams https://www.wmo.int/pages/prog/arep/wwrp/new/wwosc/documents/Williams_CAT.pdf
And conversations with the Author suggest that the Arctic Amplification is a shallow low level feature and that there is stratospheric cooling going on at Jet stream level caused by the raise CO2 IR emission level. This maintains or increases the thermal gradient driving the upper level winds and potentially increasing the Jet stream flow if strat cooling continues. The only flaw with this theory is that observed stratospheric temperatures have not cooled since 1995 after a period of major cooling from 1960 to the early 1990s.

Chuck Wiese
Reply to  pbweather
March 21, 2016 9:40 pm

pbweather: The article you reference is sheer nonsense. It is hard to believe that this sort of crap could ever be taken seriously. You have got to be kidding.

March 13, 2016 6:52 pm

“In other words, the higher the wave speed compared to the zonal current, the shorter the wave lengths.”
We don’t normally think this way outside of the electromagnetic spectrum, but perhaps we should. Rossby was very well aware that in waveforms energy is generally proportional to wavelength and the inverse of amplitude.
Intuition lies to us. We think of huge ocean waves as having enormous power. Of course they will kick your butt at Mavericks, but go back and look at the videos of the tsunami in Japan. No Hollywood cresting wave. Just a low amplitude SOB that would not quit.
So it is with the planet. The greater the internal energy, the smaller the equatorial/polar gradient, the shorter the wavelength, the lesser the amplitude:comment image
Leif has challenged me to come up with an equation for this concept. If it ever happens, it will be a 3-5 term equation you could actually describe in language. Albert so had it right. Nothing of any importance cannot be explained to a barmaid.

Reply to  gymnosperm
March 14, 2016 1:45 am

Rossby was very well aware that in waveforms energy is generally proportional to wavelength and the inverse of amplitude.

The way you say it a wave with zero amplitude should have infinite energy. You have it backward. It should be:

# In the classical wave theory, energy of a wave doesn’t depend on the frequency of the wave. However, the energy of individual photons in a beam is determined by the frequency of the beam.
# Wave’s energy is directly proportional to its amplitude squared.

Reply to  commieBob
March 14, 2016 10:04 am

I’d like to see the link. It didn’t come through.

Reply to  commieBob
March 14, 2016 12:34 pm

Although I’m anti-communist, you’re correct in pointing out that wave energy is proportional to amplitude squared. It’s wave orbital velocities that are inversely proportional to wavelength.

Reply to  commieBob
March 14, 2016 2:53 pm

gymnosperm says:
March 14, 2016 at 10:04 am
I’d like to see the link. It didn’t come through.

Darn. It works for me. Anyway, here’s the URL: https://www.boundless.com/physics/textbooks/boundless-physics-textbook/waves-and-vibrations-15/wave-behavior-and-interaction-126/energy-intensity-frequency-and-amplitude-450-1130/

Tom in Florida
Reply to  gymnosperm
March 14, 2016 6:05 am

gymnosperm March 13, 2016 at 6:52 pm
“Intuition lies to us. We think of huge ocean waves as having enormous power. Of course they will kick your butt at Mavericks, but go back and look at the videos of the tsunami in Japan. No Hollywood cresting wave. Just a low amplitude SOB that would not quit.”
Not a lie, just lack of knowledge. Anyone who has been in open water with large waves knows that the water is not moving due to the passing wave. Only when the top of the wave moves faster than the bottom will the wave break, gravity takes over and causes the top of the wave to fall. That falling water is where the power is as we who have ducked under a large break will tell you. The waves at Mavericks run into a rock ledge which causes the break. The same with Pipeline in Hawaii where the waves run into the shallow reef. Tsunami have huge wavelengths and very high speed hence the power and unrelenting movement.

March 13, 2016 6:55 pm

What is missing in this and other attempts to explain the Arctic is consideration or of its history. Almost all reported observations start after 1970 when major changes in the Arctic were over and the current regime had settled in. For most of the last 2000 years almost nothing happened there except for slow cooling. But at the turn of the twentieth century, however, the Arctic suddenly started to warm for no obvious reason. Warming continued for forty years and then stopped as though being cut off by a vent in 1940. What followed was thirty years of cooling. It lasted until 1970 when warming returned. Warming continued from then on and is still going on. It is this warming they see when they talk of Arctic amplification, a very recent phenomenon. It is quite likely that the warming at the turn of the century was caused by a major rearrangement of the North Atlantic current system that caused the Gulf Stream to start carrying more warm water into the high Arctic. The cooling in mid-century would then have o be the result of a temporary return of the original flow pattern of currents. I proved this as well as the fact that there is no such thing as greenhouse warming in the Arctic. Read my paper in E&E 22(8):1069-1083 (2011)before you say anything else about the Arctic.

Reply to  Arno Arrak (@ArnoArrak)
March 14, 2016 2:53 am

Read my paper in E&E 22(8):1069-1083 (2011)before you say anything else about the Arctic.

How about a link to a version I can read for free without having to log on or accept cookies.
This class note has the annual atmospheric heat transport as 84 W/m^2. Sea ice export is 3 W/m^2. Ocean heat transport is 3 W/m^2.
Atmospheric transport accounts for more than 90% of the heat going to the arctic. The abstract for your paper says:

All observations of Arctic warming can be accounted for as consequences of these flows of warm water to the Arctic.

That would seem to require a lot more ocean transport than is conventionally accepted. Do you have some measurements to back up your hypothesis?

March 13, 2016 7:07 pm

It will take me some time to digest the math here to see if amplitude of a Rossby wave of a period being a specific fraction of around_the_world changes as jet stream winds slow due to arctic amplification (which is for real even according to one’s choice of of any major dataset for either surface or lower troposphere global temperature anomaly, any version existing in 2008 or later).
My expectation: Arctic amplification merely slightly slows Rossby waves and other weather patterns in the northern hemisphere, without change of wave amplitude. I think the wave speed will scale downward with the westerly wind speed to keep everything constant, except speeds slowing proportionately with the square root of the ratio of temperatures (absolute, in Kelvin) at appropriate latitudes and pressure level.
Also to note: The 600 mb level is not near 18,000 feet above sea level, which is the usual figure for typical of the 500 mb level. The 600 mb level is typically around 14,000 feet above sea level.

March 13, 2016 7:33 pm

Thank heavens for people such as Chuck Wiese who have the knowledge, the time and the energy to refute claims such as those made by the warmists in their pal-reviewed paper: FV (2012).
One thing we can be certain of is that Mr Wiese did not receive funding such as the authors of the paper would have done to push their warmist agenda.

Reply to  AJB
March 13, 2016 8:41 pm

Fat finger links gone amiss, try again …
or the cream whipper back in Feb …

Steve Oregon
March 13, 2016 8:20 pm

Well Chuck, no sense in sharing that with SS, they’ll ask Appell to explain it and he’ll claim you are only a weatherman and not read it. The it’s back to bozoville for the team.

March 13, 2016 8:43 pm

With its presixties references, why doesn’t Chuck just submit this slam dunk essay to the real scientific community ?

Chuck Wiese
Reply to  WTF
March 13, 2016 8:47 pm

WTF: That’s probably a good idea. It seems they have forgotten the founding principles and never refuted any of them before publishing what they do today.

Reply to  Chuck Wiese
March 13, 2016 8:56 pm

Good luck

March 13, 2016 9:07 pm

Is this a really complicated way of restating the fundamental engineering principle, that the lower the temperature differential in, the less energy can be extracted from, a given system?

Crispin in Waterloo
Reply to  Leo Smith
March 15, 2016 2:30 am

Unless you change the pressure….

March 13, 2016 9:20 pm

“The reader needs to understand that anytime we experience severe weather, it is proof that adequate COLD in the high latitudes and Arctic has been generated by the normal radiational cooling process by the earth that creates the adequate potential energy across the latitude lines to cause amplification of the jet stream waves and speeds that pushes this colder air southward to warmer latitudes that then creates the necessary temperature gradients to liberate that energy, creating storms as well as high pressure systems.”
“If the occurrence of severe weather is increasing worldwide, it is not a sign of a warming earth. It is the opposite of what climate hysteria claims, and an indication of a cooling, not warming earth.”
The above quotes from the guest post nicely summarize the math involved.

Reply to  Chad Jessup
March 14, 2016 12:40 am

Agreed. Two clarifying quotes. Thanks for the comment.

March 13, 2016 9:25 pm

Thank you, Mr. Wiese . . Very well done it seems to me.
(“Two terms come into the discussion which are important to define. They are the amplitude and length of a wave, called amplitude and wavelength respectively.”
I suggest instead;
*Two terms come into the discussion which are important to define. They are the height and length of a wave, called amplitude and wavelength respectively.*)

Reply to  JohnKnight
March 14, 2016 12:36 am

Why introduce a confusing parameter? Amplitude is well understood, while height could be mistaken for altitude.

Reply to  Slacko
March 14, 2016 12:52 pm

Defining amplitude as amplitude seemed unhelpful.

Crispin in Waterloo
Reply to  Slacko
March 15, 2016 2:32 am

North-South span?

Reply to  JohnKnight
March 15, 2016 2:38 pm

Perhaps “latitudinal height” ? The lingo gets tricky any way you say it, it seems to me, and I think the author does a good job here and I did not mean to imply he got it wrong.

March 13, 2016 10:23 pm

Francis and Vavrus make claims about relationships and changes in relationships based on empirical research. You have presented mathematical derivations claiming that those empirical relationships can’t be real. Which of their empirically based claims are you claiming are incorrect, and how did they go wrong in estimating/displaying them? What empirical evidence can you provide to show that your derived relationships are accurate, and that theirs are inaccurate (or that theirs are less accurate than yours)?
Why exactly should Francis and Vavrus withdraw their paper?
The fact that all of your references are old is not evidence that they are wrong, but it is not evidence that they are right either. Surely much of relevance has been discovered and published since 1959? Those references antedate Lorenz, Smale, Haken, Takens, Hirsch, Guckenheimer and many other (if not all) of the developments in nonlinear dynamical systems. Lorenz, note, specifically addressed nonperiodic flow in the atmosphere.
Francis and Vavrus say this: All data are from the NCEP/NCAR Reanalysis (NRA) [26] obtained at http://www.esrl.noaa.gov/psd/. Are there written critiques of that data set that cast doubt on their trustworthiness? I mean specific criticisms, not aspersions.

Chuck Wiese
Reply to  matthewrmarler
March 13, 2016 10:58 pm

matthewrmarier: Elizabeth Barnes from Colorado State University makes a case that this “empirical evidence” you claim Francis and Vavrus provided may very well just be a statistical anomaly in the manner in which they process and use their data:
Hope you find this useful.

Reply to  Chuck Wiese
March 14, 2016 1:49 am

Most interesting, Chuck. Laid out very clearly so that even a dumbo like me can follow, if not master.
I do think Matthew has a point, though. Would it be fair to say that the Elizabeth Barnes paper you link to is a more direct repudiation of Francis and Vavrus, and that your paper provides a basic theoretical underpinning for Barnes’ findings? I also think that your request for the withdrawal of Francis and Vavrus’ paper looks way over the top and unnecessarily confrontational. Nonetheless, nice exposition – thanks

Reply to  Chuck Wiese
March 14, 2016 8:29 am

Chuck Wiese: http://barnes.atmos.colostate.edu/FILES/MANUSCRIPTS/Barnes_2013_GRL_w_supp.pdf
thank you for the link. I downloaded and read (most of) the pdf.

Chuck Wiese
Reply to  Chuck Wiese
March 14, 2016 9:25 am

mothcatcher: Thanks for your comments. The reason why I suggest FV 2012 should be withdrawn is that their paper in fact, has no theoretical basis to conclude what they do in it, and Barnes from Colorado State believes their statistical processing of the data they used was additionally flawed. FV 2012 is wrong. Should we retain papers like this given the facts? I don’t think we should. What upsets me about papers like FV 2012 when they are in obvious error is the rampant use of these flawed research papers that then spread absolute falsehoods through media that proclaim any severe weather is linked to “climate change”, AA, or to any arctic anomaly we see deviating from a mean. And I sincerely believe research on climate in many respects is being abused and licensing media to blame nearly any severe weather or even normal winter snowstorms on nonsense like this. We see and hear this proclaimed regularly since FV 2012 published. If it is not withdrawn, would the media stop using it? Probably not. I had contacted the authors of FV 2012 after they published and asked them to clarify their paper and show where basics like this are wrong. Vavrus never responded and Francis never provided the science that would refute any basics like this. From my point of view, their work should have never been published. It is misleading and wrong. These adjectives should not be the kind that describe work originating in academia.

Reply to  Chuck Wiese
March 14, 2016 10:24 am

Chuck Wiese: this “empirical evidence” you claim Francis and Vavrus provided may very well just be a statistical anomaly in the manner in which they process and use their data:
Yes, it may be.
I think it was good of GRL to publish the paper by Francis and Vavrus and then good of GRL to publish the paper by Barnes. Perhaps after more analysis and data collection, a particular statistical method will be shown to be best. Is there a mathematical theory whose consequences are confirmed by the data? With a showing of the closeness of data to model prediction?

Chuck Wiese
Reply to  Chuck Wiese
March 14, 2016 11:34 am

matthewmarler: We disagree on whether FV 2012 should have been published. You start with the assumption that your work will support accepted theories in physics unless you are out to deliberately disprove them. FV 2012 did not attempt to disprove Rossby wave physics, but in fact, they invoked Rossby wave physics to claim what they did. Then they incorrectly stated how Rossby waves behave in support of their findings. This is not how science works through any problem.
The only case in which I could see their work being published is if their results were sound ( which Barnes shows they’re not ) and they supported a case that shows Rossby is wrong. That did not happen, but in fact we find that their results were statistically processed in error giving them their results. If they had reviewed or understood Rossby waves as they should, their contradictory result would have certainly given rise to them checking their results to make sure they were not in error before publishing. Red flags should have appeared because of this but did not. All to often, papers like this are rushed through review because of the underlying message that those in academia seem anxious to perpetrate to the public. If the paper was correct, it would have been a significant finding turning Rossby physics on its ear, but in truth it is a propaganda piece supporting AGW.

Reply to  Chuck Wiese
March 14, 2016 6:05 pm

I’ll take a fairly large quote from Barnes: [9] Figures 2a and 2b show time series of JAS and
OND meridional extents calculated by the two metrics for ERA-Interim (results from NCEP and MERRA are similar). Beginning with the SeaMaxMin metric (blue curves), a significant trend emerges in JAS, with extents increasing over the past 30 years, while a large but nonsignificant trend is found in OND. Thus, the SeaMaxMin metric suggests that extents in JAS and OND have been increasing since 1980.
[10] If the SeaMaxMin metric captures the typical meridional extent of the large-scale propagating waves, then one would expect DayMaxMin to produce similar trends. Instead, very small, nonsignificant trends are seen in JAS and OND for DayMaxMin (red curves in Figures 2a and 2b). The meridional extents are smaller for DayMaxMin compared to SeaMaxMin because the extents of the isopleths
in the SeaMaxMin method are not associated with any one wave as they are for DayMaxMin, but rather, the total seasonal extrema of an isopleth. Thus, trends calculated using the SeaMaxMin metric do not reflect trends in the properties of individual propagating waves. The observed wave extents, therefore, show no trend.

Using the same metric as used by Francis and Vavrus, Barnes essentially reproduced their result. Using a different metric that, she claims, ought to provide a concordant result, she got “no change”. That does not show to me that Francis and Vavrus were wrong and had a result that ought to have been seen as wrong, but that they had an incomplete description of what they were modeling. That one metric showed a change and the other did not, would suggest to me that the two metrics are not, as assumed by Barnes, actually equivalent measures of the same thing. That a relationship based on one metric changed, and a relationship based on another metric did not change, could be the result of random variation unrelated to the rest of the measured variables, or it could be the result of having different attributes of a phenomenon change differently.
Back to your mathematical modeling, can you derive the two results reported here by Barnes? Either of them?
Granted the Francis and Vavrus paper was overhyped, but to me that is unimportant. This looks to me like something that ought to be investigated more. A common mistake in all fields of research is to report a selection of “statistically significant” results and suppress or withhold reporting of some related “non-statistically significant” results. If that is what Francis and Vavrus did, then they ought to retract their paper, or at least publish a corrigendum. Otherwise, it looks to me like they have stimulated a possibly fruitful debate.
fwiw (not much, I expect) I am skeptical of claims that increased CO2 (e.g. doubling the concentration) will produce big climate changes (e.g. increase in global mean temp more than 1C), and I am skeptical of claims that any of the consequences will be bad. I have written as much here at WUWT before.

Crispin in Waterloo
Reply to  Chuck Wiese
March 15, 2016 3:11 am

Re your quote from Barnes.
What I see is that FV 2012 has made a claim for the general with some ‘seasonal’ supporting evidence, and stated that it applies to the particular. When Barnes checked for particular evidence, it was not found. The error is for FV to have claimed that individual weather events are driven by something for which they have only ‘seasonal evidence’.
Your comment is that they used ’empirical evidence’ but that evidence is not matched to the ‘weather’ claims.
Is my understanding correct? Thanks.

Reply to  Chuck Wiese
March 15, 2016 10:51 am

Crispin in Waterloo: When Barnes checked for particular evidence, it was not found.
If I understand the papers and the long extract that I quoted, when Barnes used the metric that F&V used, she found the result that they found. When she used a different metric, she found no change in the waves being studied (more properly, she found that it was not statistically significant). It could be a reporting bias by F&V; it could be that Barnes was stimulated by F&V to do analyses she would not otherwise have thought of; it could be that the waves are not well-enough understood to come up with absolutely reliable measurements on them; it could be that there is too much additional random variation to clearly distinguish the alternatives; and you can think of more possibilities.
It seems to me that an appropriate response by F&V would be to repeat Barnes’ extended analyses, and see if they replicate the results. Only if F&V had already performed the analyses that Barnes performed, and then withheld the results (a reporting bias that, as I wrote, is very common in all sciences), might they be obligated to publish a retraction. If the disparity found by Barnes is reliable (reproducible, etc), then F&V have reported a result that requires explanation, along with the extensions found by Barnes.

Chuck Wiese
Reply to  matthewrmarler
March 14, 2016 11:56 pm

matthewmarler: You’re off in the weeds now. The authors invoked the use of Rossby physics to claim what they did. I just proved their use of Rossby physics was wrong. You asked if there was evidence showing what they presented other than what I did also pointed to error. I referenced the statistical flaws found by Barnes.
But in the end, none of that matters if the physics they claim representing Rossby are wrong or misinterpreted. The statistical arguments are of a secondary importance and if their results were valid, their next step would have been to model the replacement physics to Rossby to demonstrate he had it wrong. They never did this, but rather invoked the claims and use of his physics in error.
And I can assure you, Rossby physics are so fundamental in atmospheric science that they will never be disproven as long as the earth remains an oblate spheroid geometrically and rotates with an angular velocity of 7.292 x 10^-5 s-1.

Reply to  Chuck Wiese
March 15, 2016 11:22 am

Chuck Wiese: I referenced the statistical flaws found by Barnes.
The statistical flaw supported by Barnes’ paper was at worst a reporting bias (of a very common type), more likely that F&V considered only the one metric. Using that metric, Barnes reproduced their result. So you have a result using one metric, a different result using a different metric.
“Rossby physics” may be “fundamental”, but the derivations of results depend on simplifications that introduce approximation errors that accumulate. When empirical results (and Barnes replicated one of them) contradict the results of mathematical derivations, there is often something to be learned. And there is often too much random variation in the empirical results to conclude that there is any contradiction at all.
The statistical arguments are of a secondary importance and if their results were valid, their next step would have been to model the replacement physics to Rossby to demonstrate he had it wrong.
As Barnes wrote, one of F&Vs results was reproducible. Should future research substantiate that it is a reliable result, then someone I hope will be inspired to model it, and model the disparity with the other results. If F&Vs results had been valid, their next step would have been to publish them, just as they did. Now everyone is free to investigate whether they be valid.
If, as a general rule, nothing could be published except it agree with a priori derivations and calculations, it would be hard to learn anything new.
Note, however, I am not claiming that you are wrong on the science. I am claiming that you have not made much of a case that the F&V paper ought to be withdrawn.

March 13, 2016 10:42 pm

My instinct is to accept this article at face value because I’ve long argued that a warmer world is a more tranquil world, that reduced temperature gradients must lead to reduced extreme weather. Indeed, that’s exactly what we see in the ACE data. I note, in fact, that the “more extreme weather events” meme was originally about more intense and frequent hurricanes due to there being more energy in the system. When this argument fell apart due to both theoretical physics and observational data suggesting that lower temperature gradients resulted in the exact opposite, the warmist argument morphed (as it always does) into discussions of Rossby waves and so on. When the argument morphs in response to falsification, I always treat the new argument with a healthy amount of suspicion.
That said, I know what confirmation bias is. I’d really, Really, REALLY like this article to be true, but that fact is that the math is over my head by several fathoms. I hate to appeal to authority, but can the heavy weights comment?
RGB? Rud? Richardscourtney? Ira? etc?

March 13, 2016 11:46 pm

it takes a huge amount of math to prove that water is wet.
sometimes the qualitative explanation makes the point. less temperature differential = less turbulent weather.

March 13, 2016 11:59 pm

I want to leave a comment under Tips & Notes, but my old PC has trouble loading the page. (2100+ comments – possibly overdue for a flush.)
Change the calendar date to March 10 to see the March 9 solar eclipse from an awesome new perspective – 970,000 miles away.
direct link to one of the images

March 14, 2016 12:00 am

It’s nice to see some math but if one does small signal linearizations to find a solution, shouldn’t one at least argue that the effect isn’t contained in the non-linear terms?

March 14, 2016 2:06 am

As usual, this just sailed through ‘peer review’ without touching the sides, as it was totally in line with requirements and expectations. Perhaps now, someone will ask a few better questions of it.

Martin A
March 14, 2016 2:18 am

I’m sorry but, for me, climate modeling is a discredited field. No matter who does it nor what the results.

richard verney
March 14, 2016 2:24 am

I do not consider that we know or understand enough about climate and weather, to know what would happen.
All one can say is that irrespective as to whether the MWP, Roman Warm Period, Minoan Warm Periods were global or confined to the NH, we know that the NH has been warmer in the past with less Arctic Ice, and these were times of bounty/plenty for civilisations in the NH. Since most of us live in the NH, there is no reason to suspect that if the NH were to warm and this were to lead to a loss of Arctic Ice that it would cause any significant problem for us. To the contrary, past experience would suggest that bountiful times lie ahead.
There is nothing scary about Arctic Ice loss; heck it doesn’t even lead to sea level rise, and I for one, say bring on some warming. The globe we live in is way too cold, and also CO2 is way too low..

March 14, 2016 2:25 am

Did you try to submit a reply or comment to the original paper? I would expect such to be much more succinct than what you’ve presented here, since you supposedly would not have to educate your “peers” on so much of the basics.

Chuck Wiese
Reply to  blcjr
March 14, 2016 9:33 am

blcjr: I contacted the authors of FV 2012 right after they published and asked them to clarify their paper as described above to mothcather. No success there.

Reply to  Chuck Wiese
March 14, 2016 11:39 am

Thanks for the reply. In my experience, the normal thing to do in a case like is to contact the editors, not the authors. I.e., write up your concerns as a short “Reply to…” or “Comment on…” and submit it to the editors. The comment/reply then goes to someone independent of the authors. If the independent reviewers think your reply/comment has merit, it gets published, and the original authors typically get to reply. Authors themselves are not likely to independently acknowledge that your criticisms have any merit. They have to be forced to do so.

Reply to  englandrichard
March 15, 2016 8:30 am

It’s very evident from NSIDC that Arctic sea ice is declining rapidly. Why look at map overlays instead of actual satellite data?

son of mulder
March 14, 2016 3:00 am

Is this article the long way of saying that when warm air rises in the equatorial regions and the Arctic is warmer then the risen warm air doesn’t move quite so far north on average so it doesn’t move quite so close to the axis of the earth’s spin and hence, like the ballerina not pulling her arms in quite so far, it doesn’t move so quickly like she doesn’t spin so quickly?

March 14, 2016 3:04 am

Chuck Wiese
“FV (2012) claims a measured decrease in the zonal or west to east wind component due to “arctic amplification” (AA) would increase jet stream meandering”
It’s a meridional jet stream that causes Arctic warming, driven by increased negative AO/NAO episodes. AGW doesn’t even fit the bill on the Arctic warming, as increased GHG’s should increase positive AO/NAO. Arctic Amplification is bunkum, increased climate forcing will cool the Arctic in an interglacial climate, it’s warming since the mid 1990’s must be due to declining solar causing increased negative AO/NAO.

March 14, 2016 4:23 am

Is the last few months global temperature rise ‘exceptionally extraordinary’ ?
GISS data says: ‘far from it’ being either the exceptional or extraordinary.

Reply to  vukcevic
March 14, 2016 5:14 am

Nice to see someone looking into seasonal differences. But there is so much noise in that kind of data that it is often tricky to see any trees in the forest. Here’s a case where it actually shows something interesting:
Blue corresponds to what you are plotting, but just for the satellite era using UAH v. 6.5. Red is the seasonal difference for the natural log of the Mauna Lao CO2 series. Notice anything? (1) One can clearly see events in the temperature series such as the 97-98 Super El Nino, and the Mt. Pinatubo cooling a few years before. (2) but one can also see that the seasonal difference in CO2 growth LAGS the seasonal difference in temperature. Expect to see some spurts in CO2 levels in the atmosphere from the recent spikes in global temperature from the current El Nino.
Back to your plot, you should look at what kind of trend you get through your seasonal differences (the blue series). Is it greater in the second half of the series, compared to the first half? Is the difference statistically significant? Inquiring minds should want to know.

Andrew Kerber
March 14, 2016 5:34 am

It is interesting that you reach the same conclusion with a deep understanding of weather that I had reached with a basic understanding of physics. MY thoughts had gone like this: extreme weather is basically a release of energy. One of the things we learn in college physics is that an energy release requires a change in potential energy across a boundary. Global warming theory has the world warming, but warming much more in the polar regions than the equatorial regions, resulting in overall a more constant worldwide temperature. Since there would be less temperature difference, there would be a smaller difference in potential energy across those boundary conditions, resulting in less energy release, and thus less extreme weather.

Reply to  Andrew Kerber
March 14, 2016 12:48 pm

Since the SB law requires less energy to warm a cool blackbody by a degree than a warm one, “polar amplification” need not necessarily imply proportionally increased energy. But you’re entirely correct that decreased temperature differentials between air masses result in lower entropy production, i.e., more tranquil weather.

March 14, 2016 6:48 am

You have a type in your appendix Chuck, near the very beginning.
You have the wrong equation displayed (you repeat the equation just before it)
clip_image010 = the Coriolis parameter, defined as

Chuck Wiese
Reply to  garymount
March 14, 2016 9:44 am

garymount: Thanks for pointing this out. The Coriolis parameter is f= 2(Omega) sin ( phi). I had used Latex for the first time in putting this article together. I can tell you it was frustrating sizing the text into Word that I was using, and had to redo some of this. I’m not sure yet why this term dropped out. But you’re correct , it should be where you state.

March 14, 2016 8:00 am

Simply watch the difference of circulation between summer and winter in the boreal hemisphere and how southward lower tropospheric polar air masses MPHs do descend… Notice that contrary to Francis’ assertions the resulting jet stream is still faster in winter and yet the jet is more convoluted.

March 14, 2016 10:38 am

Most of what’s been written here goes above my head. But show me a jet stream map, and then l can understand weather. Because what the jet stream maps show me is just the “classic ice age weather pattern” that l have been waiting for to turn up over the last 3 years. Over the next 5 days the blocking pattern over NW europe and low pressure over northern Russia is going to give us a insight as to what was happening in this part of the world during the ice age. My ideas about what was the weather pattern set up during the ice age, are about to be tested in the real world.

March 14, 2016 8:38 pm

Regarding Phase D in Figure 6: In my experience, at the 500 millibar level (where such things are usually depicted), and even lower aloft when a jet stream path is clearly discernable (which is a majority of the time at the 700 mb level), advanced lows poleward of the jet stream are usually more poleward than advanced highs equatorward of the jet stream. This is despite the jet stream dipping towards the equator around such lows and reaching poleward around such highs.
Even Arctic highs that come down North America’s Great Plains in the winter and that achieve high surface barometric pressure are shallow highs at first, and they come from north of the jet stream. Sometimes they achieve high surface pressure and then (usually even later) they get the jet stream taking a curvy path north of them, but by that time the surface high pressure center is usually around or south of 45-50 degrees north latitude, and at the 700 mb level either this high is only a ridge or there is a closed high south of 45 degrees north latitude.
As for lows: Much of the time, especially in winter and early spring, there is persistent low pressure around/near the Aleutian Islands and around Iceland or southern Greenland – sometimes as far west as eastern Canada. These are the “semipermanent lows”. Usually a major storm south/southwest of these two items takes a path where it gets sucked into ones of these “storm retirement areas”, or it gets sucked into another storm heading that way – ultimately, most major storms here have lowest altitude of the 700 mb level north of 50 degrees north latitude. And when a major extratropical storm around North America peaks elsewhere in terms of surface pressure or height of the 700 or 500 mb level, the lowest altitude of the 500 and even the 700 mb levels are usually around or north of 45 degrees latitude.
Please find analyses and forecast models for the 700 and 500 millibar levels to verify this on your own. One that I know of, 9-panel for the GFS forecast model for days 2 to 10 into the future, with surface (sea level extrapolated) isobars and 500 millibar heights in meters plotted by color code, is at:

March 15, 2016 11:34 pm

hard to digest article but nice read…
what i can say is that with simple life observations here in Belgium i can kinda “back this up” in a much easier way:
we are very dependant of the western circulation that is bringing us the mild winters. Now if blocking would occur more often then normal due to the temperature gradient decrease,and decrease of the arctic amplification, we would see ironicly more severe cold winters. blocking here results in the transport of continental cold airmasses from siberia.
actually these blockings didn’t occur and we had the last years even ver strong westerly currents resulting in record warm winters paired with record long positive NAO index.
in short if there would be more blocking and waves, we would see more rain spells/cold spells swaps in our winters. the more the westerly wind is “clean of rossby waves” the more “above average” our winters in fact become (and ironically the more rainy too) as then the heat (and moist) of the gulf stream is transported to land.
conclusion: longer rossby waves do explain this als longer waves mean longer time intervals to swap weather patterns here

March 17, 2016 9:22 pm

I have been studying this article with the intent of thoroughly understanding every last word of it. I have several years of calculus study under my belt so the math is not much of a problem, except I haven’t done much math for the past 6 years and might need to do some review.
In the comments for this article there are many who express a desire to have some guidance to understanding.
I am prototyping a means of study or learning by converting this article into text and code in a software development environment, a screen shot of which is below. I have started at the appendix instead of the beginning of the article. It has only taken me an hour or so to create what you see so far. The directed graph on the left was created by simply selecting items from the solution explore on the right and dragging and dropping.
I am using Test Driven Design / Development (TDD) to flesh out classes this mini project might need. So far none of the classes have any implementation, methods throw a “NotImplementedException” for now.
I create classes directly from within the test methods by using what is called “generate by usage”, whereupon you select text that represents a class type that doesn’t have any code written yet and hit ctrl + dot ( . ) and select “create class” and the code for the class gets created for you without leaving the test method.
What I am doing isn’t truly faithful to the TTD philosophy, as you usually decide on a piece of functionality to implement, whereas here I am creating a skeleton of code wrapping around a blog post.
Soon enough though there will be implementations written, using the TTD philosophy where no code is written without first there being a test class and test methods driving the development of the implementations.
Well, that’s enough for now. There are far more details of this mini project to explore to the completion of an app that has visuals, etc. For example refactoring of the code is carried out at the project progresses. ViewModels are created that defines the API for the application. A separate App that has nothing but view code for the UI, that uses the ViewModels, essentially to provide a more or less platform independent code base.
I am using the free Microsoft Visual Studio Community 2015 (update 2 RC) to produce everything you see here.comment image

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