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Comments 351 to 400 out of 529:

351. Quietman at 05:15 AM on 8 December, 2008
     Arkadiusz At least you are a scientist. Many of us posting here can't even say that. Personally I find your comments (and Patricks) quite interesting.
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352. Patrick 027 at 16:04 PM on 10 December, 2008
     Rossby Wave Wrap-up 1a. Linear superposition: For any amplitudes, multiple sets of vorticity anomalies have multiple wind fields associated with them, and each adds linearly to produce a total vorticity anomaly field with a total wind anomaly (which can be added to the basic state vorticity and wind to get total vorticity and wind). For relatively weak amplitudes (where displacments are relatively small compared to wavelength and variations in the basic state), the changes in vorticity over time due to potential vorticity advection by the wind field (which results in propagation of the vorticity anomaly patten) can also be approximated with linear superpositions of multiple vorticity anomaly patterns - each propagating in it's own way. However, the changes in potential vorticity anomalies are due to the displacements of potential vorticity contours. When displacements by all waves are along the same direction (with anomaly wind vectors purely parallel to that direction, such as occurs with vorticity waves with no variation in amplitude along the length of infinite phase lines), and with the basic state PV gradient (or at least the component parallel to the wind anomalies) constant along the same direction, and assuming constant ratio between PV anomaly and RV anomaly (it could vary due to different degrees of divergence and convergence due to ...), then the total displacement is equal to the sum of displacements of individual waves and the total resulting change of PV is the same at any x,y point, so the waves can still be linearly superimposed. However, more generally, there can be nonlinearities that arise because, 1. if the PV gradient varies along the direction of displacement, then the PV gradient can be changed at a fixed location by that displacment; an additional wave acting at the same location is no longer acting on the same PV field. 2. as the PV field is displaced by the anomaly wind, changes in the PV gradient can be produced (such as by variation in anomaly wind along phase lines) so that the change in PV produced by the next anomaly wind added are not proportional. 3. variation in amplitude of a vorticity wave along phase lines requires some closed streamlines - the anomaly wind varies in direction. PV contour displacements in one direction can alter the PV gradient in another direction. 4. Maybe some other things I haven't thought of yet. ** In particular, in the case (**CASE C1a** for future reference; **CASE C1b** will refer to a case when the PV gradient is not entirely in the y-direction) of a vorticity wave phase lines aligned in the y direction, phase propagation in the negative x direction, basic state PV gradient in the y direction, where the vorticity wave amplitude is a maximum at y = 0 and decays to 0 toward y = A and y = -A (while being symmetrical about the x axis), then, at y = 0, the anomaly wind only has components in the y direction, but away from the x axis, the anomaly wind also has x components. As the wave propagates in the negative x direction, setting aside the basic state wind, the air flows through the wave in the opposite direction at the phase propagation speed at y = 0, but the x component of the wave alternately varies the flow through the wave, causing the air to spend more time within one phase of the wave and less in the other, and affecting the resulting displacements of PV contours; the result is to sharpen and intensify the vorticity wave crests and spread out and weaken the vorticity troughs on one side of the x axis and the reverse on the other side. Obviously this effect must increase when the x component of the anomaly wind is large compared to the phase speed in the x direction. (Some analogy might be made to water gravity waves when the back-forth displacements are large in comparison to the wavelength; in which case the crests are sharp and the troughs are broad). Of course this change in wave form could modify the propagation itself... ---------- The vorticity wave has a wind wave, with components u' and v', in the x and y directions respectively; they will be (below somewhere) be refered to as the u' wave and the v' wave. 1b. wave numbers, phase speeds, group velocity: Remember that the wave numbers (k in the x direction, l in the y direction (though I've also seen l,m instead of k,l used), which add as vector components to give the wave vector (k,l)) are inversely proportional to the wavelength (1/k in the x direction, 1/l in the y direction, 1/(wave vector magnitude) in the direction of the wave vector, which is *THE* wavelength, in the direction perpendicular to phase lines). Because phase speeds are the speeds of phase lines and thus proportional to wavelength (equal to wavelength times frequency); the phase speeds don't add like vectors; but their inverses do. But if I ever refer to 'phase velocity', that might not be a correct term, but what I am refering to is the phase speed in the direction of the wave vector (perpendicular to phase lines). Group velocity is the motion of a pattern of amplitude variation of a wave (for a wave pattern with amplitude varying in space); group velocity is a vector that is the sum of group velocity components in the x and y direction (or any two orthogonal dirctions, such as parallel and perpendicular to phase lines). 2a. Linear superposition and patterns, group velocity: Consider waves of constant amplitude with wave numbers k and l. For waves with l = 0, phase lines are in the y direction. Waves with l = 0 but different k form an interference pattern with amplitude varying in the x direction. If the two k values are only slightly different, then the interference amplitude pattern moves in the x direction at the group velocity of the wave with a k equal to the average of the aforementioned two k values. This can be seen using one of the trigonometric relationships: 2 * cos(a) * cos(b) = cos(a+b) + cos(a-b) = cos(a+b) + cos(b-a) 2 * sin(a) * sin(b) = cos(a-b) - cos(a+b) = -[cos(a+b) - cos(a-b)] 2 * cos(a) * sin(b) = sin(a+b) - sin(a-b) = sin(a+b) + sin(b-a) OR, where sum = a+b and dif = a-b: 2 * cos(a) * cos(b) = cos(sum) + cos(dif) = cos(sum) + cos(-dif) 2 * sin(a) * sin(b) = cos(dif) - cos(sum) = -[cos(sum) - cos(dif)] 2 * cos(a) * sin(b) = sin(sum) - sin(dif) = sin(sum) + sin(-dif) ---------- 2b. So the linear superposition of: cos[(k+dk)*x - (w+dw)*t] and cos[(k-dx)*x - (w-dw)*t] is: 2 * cos(k*x - w*t) * cos(dk*x - dw*t) Which can be seen (assuming |dk| << |k|, |dw| << |w|) as: a wave cos(k*x - w*t), which has x-direction wavenumber k and angular frequency w, and phase speed in the x direction equal to w/k, (we expect w/k to be negative for PV gradient in the positive y direction), modulated by an 'amplitude wave': 2*cos(dk*x - dw*t), which has wavelength 2*pi/(dk), and moves in the x-direction with the x-component of group velocity dw/dk. There is no variation in the y-direction of this pattern. Call this **CASE C0** (This is nicely explained in Appendix A of Cushman-Roisin.) ---------- 2c. MORE GENERALLY, The linear superposition of: cos[(k+dk)*x + (l+dl)*y - (w+dw)*t] and cos[(k-dx)*x + (l-dl)*y - (w-dw)*t] is: 2 * cos(k*x + l*y - w*t) * cos(dk*x + dl*y - dw*t) Which can be seen (assuming |dk| << |k|, |dl| << |l|, |dw| << |w| **(actually, the last condition may not be necessary, but in the limit of small dw, dw/dk and dw/dl can be treated as partial derivatives of w as a function of k and l, in other words as the components of the gradient of w in k,l space) as: a wave cos(k*x + l*y - w*t), which has wave vector (k,l) and angular frequency w, and phase speeds in the x direction equal to w/k, and in the y direction equal to w/l; the phase lines have slope dy/dx = -k/l modulated by an 'amplitude wave': 2*cos(dk*x + dl*y - dw*t), which has wavelength 2*pi/[(dk^2 + dl^2)^(1/2)], and has ** group velocity (dw/dk,dw/dl) **. ----- 2d. **???(PS why is group velocity given as a vector with components in the x and y direction? These components are the velocities in those dimensions of a point on a 'phase line' of the 'amplitude wave' that doesn't (in this case, at least** - more generally the group velocity is the velocity of an 'amplitude region' which may not be infinitely long and straight) move along the length of the 'phase line', only perpendicular to it, keeping up with it. The reason why phase speeds in x and y directions don't add as vectors to give the 'phase velocity' is because they are not generally the components of motion of such a point on the phase line; rather they are the speeds of motion of the points that are the intersections of a phase line with a line parallel to the x-axis and then a line parallel to the y-axis. )???** ---------- 2e. When l = 0 and dk = 0, the pattern is: the linear superposition of: cos[k*x + dl*y - (w+dw)*t] and cos[k*x - dl*y - (w-dw)*t] which is: 2 * cos(k*x - w*t) * cos(dl*y - dw*t) Which can be seen (assuming |dl| << |l|, |dw| << |w|) as: a wave cos(k*x - w*t), which has wave vector (k,0) and angular frequency w, and phase speeds in the x direction equal to w/k, modulated by an 'amplitude wave': 2*cos(dl*y - dw*t), which has wave vector (0,dl), and has group velocity (0,dw/dl). Call this **CASE C2** **CASE C2a**: Now, in this case, if the basic state PV gradient is in the y-direction (the 'default setting' for this overall discussion), then for Rossby waves, dw/dl should be zero at l=0. There is 'amplitude wave' propagation in the y direction; the phases propagate in the negative x direction. **CASE C2b**: But if there is a basic state PV gradient in the x-direction as well, then dw/dl can/will be nonzero and there will be 'amplitude propagation' in the y direction. Physically, without reference to the motion of the linearly-superimposed components that create the pattern (though it could be understood that way as well for small amplitudes), the reason for this is qualitatively the same as **PART OF** the reason for 'amplitude propagation' in the same direction in **CASE C1b**, which is that, as the amplitude varies along phase lines, there are vorticity maxima and minima that are maxima and minima in both dimensions x and y. This means that the anomaly wind streamlines are closed loops; not only is there a v' wave but also a u' wave. u' is positive where the vorticity anomaly decreases in y and is negative where the vorticity anomaly increases in y. When the basic state PV contours are not parallel to the x-axis, this x-component of the wind displaces those contours so as to propagate variations in amplitude along the phase lines. If the x-component of the PV gradient is toward positive x, then this along-phase-line propagation is toward positive y; it is in the negative y direction for a PV gradient x-component in the negative x direction. Notice if we realign the axes with the PV gradient then the phase lines are tilted and this describes the component of group velcocity parallel to phase lines. In **CASE C2b** in particular, the u' wave has crests and troughs at the nodes of the 'amplitude wave', just as v' has crests and troughs at the nodes of the vorticity wave that occur at discrete x values (at any one time). The nodes of the wind waves pass through the vorticity maxima and minima - which is a quick way to make the judgement that the vorticity maxima and minima keep the same amplitude as they propagate along phase lines ('amplitude propagation') while the phase lines also propagate. In **CASE C0**, the group velocity is perpendicular to phase lines and generally of different magnitude and/or direction as phase propagation, which requires that, following individual phase lines, the vorticity maxima and minima grow and shrink, and reverse sign; this requires that the wind wave nodes not pass through the vorticity wave crests and troughs except at the 'crests' and 'troughs' of the 'amplitude wave'. Notice that in **CASE C2**, rather than refering to one wave field modified by 'amplitude waves', the pattern instead can be described as a checkerboard pattern of rectangularly shaped vorticity wave phases (with u' and v' wave phases also rectangular). The along-phase line 'amplitude propagation' can also be desribed as the y-component of phase propagation of this checkerboard pattern. This becomes more obvious if |dl| is not much smaller than |k| (in which case it doesn't make sense to keep the 'd' - call it l instead of dl). What is also true is that the the identification of the 'amplitude wave' can be assigned to the other part of this pattern (the part with wavenumber k), which becomes more obvious when |l| is larger than |k|. Another point that is interesting, which applies to all cases **CASE C1** and **CASE C2**, is that because -du'/dy makes a contribution to vorticity anomalies (generally with the same sign as dv/dx for the cases as described) , dv'/dx will be smaller for the same vorticity anomalies, which means that, for the same vorticity wave amplitude and wavelength in the x direction, v' wave will have smaller magnitude than otherwise, more so as the magnitude of dl increases (as the wavelength in the y direction shrinks, or more generally, as the spatial scale of the vorticity variations in the y-direction shrinks, for a given vorticity variation). (and the same for u' when the x-direction wavelength shrinks for a given y-direction wavelength, etc.) This means that for the conditions so far and for the same ratio between anomaly PV and anomaly vorticity, the propagation in the x direction will be slower for given vorticity variation over shorter y direction distances (and so on for y-direction propagation with shorter x direction distances). to be continued just a bit more... _______
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353. Patrick 027 at 16:08 PM on 10 December, 2008
     PS a little unsure of my section 2d above.
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354. Philippe Chantreau at 18:46 PM on 10 December, 2008
     I wouldn't hold it against you Patrick... Quietman, you may be right about Arkadiusz, or not. I am still unimpressed with Beck, especially his fantasy graph on Dansgaard-Oeshger cycles. Has he withdrawn that piece of work from his web site? Has he explained what the change in the x axis mean? Arkadiusz, what degree have you earned from which institution and what are your publications? Those are the things that will tell me if you are a "scientist", as Quietman states. I do applied science too (in another area), and would certainly not consider myself a "scientist."
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355. Patrick 027 at 05:28 AM on 11 December, 2008
     Correction sec.2e.: **CASE C2a**: Now, in this case, if the basic state PV gradient is in the y-direction (the 'default setting' for this overall discussion), then for Rossby waves, dw/dl should be zero at l=0. There is *** N O T *** 'amplitude wave' propagation in the y direction; the phases propagate in the negative x direction.
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356. Patrick 027 at 16:13 PM on 11 December, 2008
     ... In **CASE C2**, the checkerboard pattern - the nodes of the vorticity wave, u' wave, and v' wave, form a set of rectangles for each wave. Contours of wave values thus are nearly rectangular near the nodes but become more rounded toward the centers of the rectangles. In this checkerboard pattern, parallel to either set of parallel nodes, wave values vary sinusoidally. In the x direction, the u' and vorticity waves are in phase and 180 deg out of phase (depending on y), while they are 90 deg or 270 deg out of phase in the y direction. In the y direction, the v' and voriticity waves are in phase and 180 deg out of phase (depending on x), while they are 90 deg or 270 deg out of phase in the x direction. Because of the constant proportionality in the x direction of u' with vorticity and in the y direction of v' with vorticity, the propagation in the y direction does not vary with x and the propagation in the x direction does not vary with y. (The propagation of the wave of course occurs as u' and v' act across the PV gradient to increase PV where the PV anomaly gradient is in one direction and decrease the PV where a component of the PV anomaly gradient is in the other direction; a maximum moves toward where values increase and away from where values decrease, a minimum moves toward decreasing values and away from increasing values; the derivative in space of a sinusoidal waveform is another sinusoidal waveform either 90 or 270 deg (depending on view point) out of phase; a moving sinusoidal waveform produces variations at fixed locations that are sinusoidal in time, etc, so propagation of a otherwise unchanging wave pattern will have rate of change of PV, RV, u', v', etc, 90 or 270 deg out of phase (in the direction(s) of propagation) from the wave pattern of the instantaneous values of PV, RV, u', v', etc, respectively, and with the amplitude of the time derivative wave in proportion to the amplitude of the instantaneous value wave - a proportion that can vary in the direction of propagation but does not vary in the perpendicular direction). -- But what about **CASE C1**, or more generally, when a wave does not form an infinite pattern but has only nonzero amplitude (or amplitude above OR below some threshold) in a single limited region that is not part of a repeating sinusoidal variation (or some linear combination of those) of amplitude? (PS this does not describe any propagation of the checkerboard pattern; the modulation of amplitude of one wave by another sinusoidal function has a group velocity but this is not the group velocity of the whole pattern; to illustrate such a group velocity, one must multiply the whole wave form by yet some other amplitude-modulating function, etc...) Take **CASE C1** for example. The vorticity wave is a wave train along the x-axis, symmetric about y, that is nonzero only over some finite range of y. If the basic state PV gradient is not parallel to either axis, as in **CASE C1b**, then the wave is tilted and the 'amplitude wave' (better term - the wave envelope) will move with some group velocity in y that is a function of the wavelength in x (and therefore a function of wavenumber k), where the group velocity y component is precisely the derivative dw/dl for the k value - w is a function of (k,l) for Rossby waves and waves in general; this is how group velocity can be defined for a wave that is not part of a specific interference pattern of specific waves). In **CASE C1a**, the y component of group velocity is 0; the wave envelope doesn't propagate in y. The physical explanation is qualitatively the same as in **CASE C2(b and a, respectively)**. A narrow wave envelope should slow propagation in the x direction for the same reason that k affects propagation in y and l affects propagation in x for the checkerboard pattern. But that is only part of the story. In both **C1a** and **C1b**, as in **C2**, there are closed wave streamlines around the wave vorticity maxima and minima, as there are u' and v' waves. But in **C2**, both u' and v' reach maxima and minima along the nodes of the vorticity wave, and there values go toward zero approaching the next vorticity maxima or minima. In **C1**, in the y direction, there is not other vorticity maxima or minima. Instead, u' and v' values must decay toward zero going away from the center of the wave envelope - Refering back to how the wind field can be determined from the vorticity field, the length scale of this decay to zero will increase with increasing wavelength in the x-direction (note consequences for group velocity, this is somewhat** qualitatively similar to how k affects propagation in y and l affects propagation in x for the checkboard pattern, though perhaps for additional reasons). Outside the vorticity wave envelope, this wind field must also be irrotational; this works because while -du'/dy must reverse sign (in the y direction) in order for u' to decay to zero, dv'/dx can keep the same sign out to large |y| (though it approaches 0 as v' approaches 0). One can get a qualitative handle on this (and some of the other issues discussed above, including group velocity of tilted waves) by considering each phase line as a string of circular vorticies (remember that the wind speed of each is inversely proportional to distance from the center of each). Each contributes to the wind field v' in between vorticity maxima and minima phase lines, and not just at points on the same y value as the vortex; hence, v' is larger at any one y value due to voriticity at other y values (though the vorticity at the same y will have the greatest effect). Meanwhile, the u' fields of any pair of vortices also adds to increase total u' outside the pair but the values partly cancel in between the pair (with the total u' from that pair being of the sign of the u' from the stronger of the two vorticities). And so on... vorticity phase lines of constant amplitude over y has constant v' over y as well, zero u', and if only one line of vorticies, v' would be constant over x on either side of the line. But variation of vorticity amplitude along a phase line allows nonzero u' (as described above for both cases **C1** and **C2**, thus allowing some propagation in the y direction if there is a PV gradient in x ----- [PS another way to look at that: suppose instead of holding x and y oriented to the wave structure, hold x and y so that the PV gradient is just in y, but the wave is tilted. The wind field of each vortex will, by displacing PV contours, increase PV on one side and decrease PV on the other, with zero PV change along a line in the y direction centered on the vortex. In a tilted wave, the vortices along a phase line (along a vorticity wave crest or trough) lie in each other's PV changing regions. Where their is constant amplitude along the phase line or where there is an amplitude minimum or maximum (constant amplitude at that point), provided symmetry about that maximum or minimum, then the effects cancel and there is zero PV change at that point, but where vorticity is not symmetric along the phase line about a point, or generally where vorticity changes along the phase line or changes sign, the effects at the location of one vortex by the other vortices can be or will be unbalanced, so that PV is changing at that point along the vorticity crest or trough, hence there can be or will be amplitude propagation in the direction parallel to phase lines]. -----), and variation of v', but unlike **C2**, where the v' is kept proportional to vorticity over y, for **C1**, the parts of the wave near y=0 may generally have less v' per vorticity as there is on the edges of the vorticity wave envelope - the stronger vorticies have proportionately weaker v' at the same y due to the weaker vortices at other y values, while the weaker vortices have stronger v'. Furthermore, outside of the vorticity wave itself, there is still v' (and u') from the vortices within the wave envelope (that their magnitudes decrease faster with distance from the vorticity wave envelope for shorter wavelengths in the x direction (higher k) can be seen as a consequence of the wind field at any one point depending less on the finer details at some distance). This means that, initially, phase propagation speeds (in the x direction) are slower in the center of the wave envelope as they are on the outskirts. This means the waves bend into V or U shapes (and the wave envelope expands due to the extent of v' (and u' for **C1b**)). The wave is tilted relative to (x,y), above and below y=0, in opposite directions. This bending then allows for amplitude propagation away from the center (where the amplitude falls) and toward the wave envelope edges; for **C1a**, this is symmetrical about y=0, for **C1b**, the variation of basic state PV in the x direction can introduce some asymmetry - the tilts relative to the PV gradient direction won't be equal and opposite; conceivably they might be the same sign; they can't have the same magnitude, though; ----- (**The component of Group velocity parallel to phase lines - amplitude propagation along phase lines - is fastest at intermediate tilts between phase lines being aligned with the PV gradient and being perpendicular to it**). This could be seen as two wavetrains of equal and opposite phase tilts relative to x,y directions, but with wave envelope aligned in the same direction, that were initially linearly superimposed, but then seperated as each had it's own group velocity (and possibly different phase phase speeds, if the PV gradient is not parallel to y). However, each of these wavetrains can be expected to undergo the same process (modified by different phase tilts, etc.) as occured with the original wavetrain. Alternatively, depending on wave envelope form, there might never be complete seperation (thought the amplitude at the center will continually decrease) - the wave might continue to spread with a range of group velocities (corresponding to those group velocities of all linearly superimposed component waves) with the phase lines curving into U shapes. Another way of veiwing this is to consider a string of vorticies of alternating sign (representing the wave train, aligned with the wave envelope); rather than consider the motion of the vorticities, one can think of it as vorticities that are not moving but with each generation of vorticity phase lines continually producing a new generation of new vorticity crests and troughs that are 90 or 270 deg out of phase from the parent generation. The total wave propagates because the third generation is 180 degrees out of phase from the first, as is the forth from the second, etc, so that they cancel each other. But with amplitude confined to a wave envelope centered at y=0, any generation will produce a next generation that is more spread out in y and has lower amplitude at y=0. Thus the third generation does not completely cancel the first at y = 0, but that allows a portion of the first generation to continue to act to produce an additional second generation (generation 2B ?), and so on... so that the first generation might eventually be canceled out, but by that point ... etc...
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357. Patrick 027 at 16:17 PM on 12 December, 2008
     CORRECTION " (**The component of Group velocity parallel to phase lines - amplitude propagation along phase lines - is fastest at intermediate tilts between phase lines being aligned with the PV gradient and being perpendicular to it**). " Actually, that's for the group velocity y-component with basic state PV gradient parallel to y. The fastest group velocity component parallel to phase lines occurs when the phase lines become parallel to basic state PV contours. Of course, for finite-width wave envelopes, this should be equal to phase propagation of the same wave described instead as having phase lines aligned with the PV gradient, with amplitude varying sinusoidally along phase lines... etc. ------------ A wave envelope can be limited in multiple dimensions. In **C0** in was limited in the x direction. One could consider the case of a wave envelope limited in both x and y, in which case aspects of **C0** and **C1** would be combined; there could/would be group velocity in both x and y; new phase lines would grow on the outskirts in the x direction (in the direction of group velocity minus phase motion - if the wave envelope is not spreading out too quickly, then old phase lines will decay on the other side of the wave envelope). ---------- Group velocity: THE group velocity of a wave is determined by the frequency as a function of the spectrum - specifically, it is in (x,y) space equal to the gradient of w in wavenumber space (k,l) (and so on for three dimensional waves, etc.) - thus it has components that are partial derivatives dw/dk and dw/dl. ----- (I'm using d for partial derivatives here but partial derivatives are written with a "del" symbol (not the same as the gradient operator symbol, which I believe is called "Del", which is an upside down capital greek letter Delta; the del symbol looks a little like the lowercase greek letter delta, but is smoother - it looks like a backwards '6'. I also used 'd', as in 'dw','dk','dl' above, as values (representing a difference in w, k, or l) that may or may not be infinitisimal in size. The propper symbol to use in that case is the lowercase delta, or particularly for sizable differences, capital Delta. I'm going to continue to just use dw, dk, and dl here, though.) ----- But in order to actually see amplitude propagation (wave envelope propagation) at the group velocity, there must be variation in amplitude. This can be produced by linear superposition of additional waves that or only infinitesimally different parts of the spectrum. In that case, amplitude variations are very spread out in space and the group velocity of the interference pattern is about the same as (dw/dk,dw/dl). However, as the amplitude variations become more concentrated in space, the ratios of the differences of w to the differences of k and l between wave pairs won't be exactly the same as the derivatives dw/dk and dw/dl for each linearly-superimposed component - and each component may have different dw/dk and dw/dl for it's own k and l. While a wave envelope will propagate with some average or effective group velocity, it will also tend to spread and weaken (or contract and intensify up to a point and then spread and weaken - one or the other might happen in one direction while the opposite happens in another direction) and the phase lines may take on different tilts in different parts of the wave envelope, which might then be described by multiple overlapping wave envelopes, etc..., as there are a range of group velocities present. While wave envelope propagation perpendicular to phase lines can always be seen as being at a group velocity component in that direction, the group velocity along phase lines may lose any meaningful distinction with phase propagation, as in the checkerboard pattern example; this suggests (at least for Rossby waves) that the group velocity component parallel to phase lines will get smaller when the wave envelope wavelength in that direction get's smaller, just as phase speed is smaller for smaller wavelengths, and as described for the checkerboard pattern, both propagation of phases or along phases each vary qualitatively the same way with wavelengths in both directions.
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358. Patrick 027 at 15:38 PM on 14 December, 2008
     I figured out the error I made in sections 2c and 2d in comment 352 (PS and I may have made this error earlier as well when discussing group velocity). Recap: The linear superposition of two waves of equal amplitude, with average wave vector (k,l) and average angular frequency w, with wave vector difference between them 2*(dk,dl) and difference of angular frequency = 2dw: cos[(k+dk)*x + (l+dl)*y - (w+dw)*t] + cos[(k-dx)*x + (l-dl)*y - (w-dw)*t] = 2 * cos(k*x + l*y - w*t) * cos(dk*x + dl*y - dw*t) which can be seen as a wave with wave vector (k,l) and angular frequency w (with phase lines with slope dy/dx = -k/l) modulated by a sinusoidal wave envelope with 'amplitude wave vector' (dk,dl) and frequency dw (with 'amplitude phase lines' with slope dy/dx = -dk/dl). The magnitude of the wave vector (k,l) is M = (k^2+l^2)^(1/2), and the wavelength of the wave with that wave vector, measured in the direction of the wave vector (perpendicular to phase lines) is 2*pi/M (and so on for 'amplitude wave vector' (dk,dl) ). Now, here's the error: "dw/dk and dw/dl can be treated as partial derivatives of w as a function of k and l, in other words as the components of the gradient of w in k,l space" To borrow a phrase from Ted Stevens, NO! As with w/k, w/l, and w/M (phase speeds in x direction, y direction, and direction of wave vector ("THE" phase speed))for the wave with wave vector (k,l), wave vector magnitude M, dw/dk, dw/dl, and dw/dn are the phase speeds of the wave envelope in the x direction, y direction, and direction of 'amplitude wave vector' (dk,dl), respectively, where dn = (dk^2+dl^2)^(1/2) is the magnitude of half the vector difference between the wave vectors of the two linearly superimposed waves. As with the wave with wave vector (k,l), the phase speeds dw/dk and dw/dl do NOT add as vector components to give a vector in the direction (dk,dl) with magnitude dw/dn - it won't generally have that magnitude nor will it have that direction. The inverses, however, do have the relationship of vector components; (dk/dw,dl/dw) = dn/dw *(dk,dl)/dn, where (dk,dl)/dn is the unit vector in the direction of (dk,dl). So what is the group velocity, and what are it's vector components? This is where it would have helped me avoid a mistake if I had been using the 'del' symbol for partial derivatives; it is important to differentiate (pun intended) among the different kinds of derivatives. The group velocity for a wave with wave vector (k,l) is the gradient of w over wave vector space at the value (k,l), with the component in the k direction being the component of group velocity in the x direction, and so on for l and y, or for any other set of orthogonal components that could be used (the component in the direction of a vector (a,b) in k,l space would be the component of group velocity in that direction in x,y space. The gradient is a vector with components equal to partial derivatives; group velocity = (cgx,cgy) = [del(w)/del(k) , del(w)/del(l)], where del(a)/del(q) is the partial derivative of a with respect to q. The key is that del(w)/del(k) is NOT generally equal to dw/dk, even in the limit of infinitesimal dw and dk such that dw/dk is equal to a derivative. This is because dw is the total difference in w over a specific distance in k,l space in a specific direction. Only if that direction were only in k - that is, if dl = 0, would then del(w)/del(k) be equal to dw/dk (in the limit of infinitesimal dk). More generally, in the limit of infinitesimal differences, dw = del(w)/del(k) * dk + del(w)/del(l) * dl; dw/dn = del(w)/del(k) * dk/dn + del(w)/del(l) * dl/dn That is - only part of dw is due to dk, etc (and so on if we were discussing three-dimensional waves). dk and dl are components of the vector with magnitude dn in the direction of (dk,dl)/dn, which is of course that vector, (dk,dl). dw/dn IS the component of group velocity in that direction, because it is equal to the partial derivative of w in that direction, given how dn was defined. This is the phase speed of the wave envelope in the direction of its wave vector. It has vector components in x and in y, which are equal to dw/dn * dk/dn and dw/dn * dl/dn, respectively. These are not components of the total group velocity, however, because they are components of a component vector. One would have to add to each component the component of the other component of the group velocity to find the total group velocity components in x and y. Say that n is the distance perpendicular to s, that is, perpendicular to the 'wave envelope vector' (dk,dl); such a vector would be in the direction (dl,-dk) (90 deg to the right) or opposite that, (-dl,dk) (90 deg to the left). Let's have it be the second one, so that coordinates s,n are a rotation of x,y. In that case, del(w)/del(s) is the component of group velocity in the s direction, which adds with the vector in the n direction with magnitude dw/dn to give the total group velocity. Of course one won't observe such a group velocity for a wave envelope that does not vary in s. More generally, if a wave envelope has a rectangular checkerboard pattern that is described with two sets of 'phase lines' intersecting at right angles (as the nodal lines would do), then the two 'phase speeds', each of one set of wave envelope 'phase lines' in the direction perpendendicular to those 'phase lines', form a complete set of components of the group velocity. The group velocity will be the velocity of the checkerboard pattern, which can be defined by the motions of points (such as node intersections), as opposed to lines. ... Suppose we instead take s and n as coordinates locally defined in k,l space as being perpendicular and parallel to the wave vector (k,l) (with magnitude M) of a wave, respectively (or parallel and perpendiclar to phase lines of that wave, respectively), with n 90 deg to the left of s, so that n is in the direction of (k,l) and s is in the direction of (l,-k). Then group velocity can be described with components: cgn = del(w)/del(n) in the direction of phase propagation, which is in the direction of the wave vector and cgs = del(w)/del(s) in the direction along phase lines, to the right of phase propagation. The vectors (cgs,cgn) and (cgx,cgy) are the same vector, the group velocity, described in different coordinates - s,n and x,y, respectively. Using dot products with unit vectors that define s and n directions ( (l,-k)/M and (k,l)/M, respectively), cgs = (cgx,cgy) "dot" (l,-k)/M = 1/M * (l*cgx - k*cgy) cgn = (cgx,cgy) "dot" (k,l)/M = 1/M * (k*cgx + l*cgy) which implies that l/M is the cosine of the angle between y and n directions and the cosine of the angle between x and s directions, while k/M is the cosine of the angle between x and n directions and the negative of the cosine of the angle between y and s directions. (which can be confirmed geometrically).
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359. Patrick 027 at 16:00 PM on 14 December, 2008
     CORRECTION of a paragraph above, with changes noted ***: "Say that s*** is the distance perpendicular to n****, that is, perpendicular to the 'wave envelope vector' (dk,dl); such a vector would be in the direction (dl,-dk) (90 deg to the right) or opposite that, (-dl,dk) (90 deg to the left). Let's have it be the FIRST*** one, so that coordinates s,n are a rotation of x,y. "
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360. Patrick 027 at 16:13 PM on 14 December, 2008
     ..."the negative of the cosine of the angle between y and s directions." It implies that the angle between y and s directions is an 180 deg more or less than the angle between x and n directions, which can be confirmed geometrically. The angle between x and s directions is the same as the angle between y and n directions, but the angle between x and n directions is the angle between the y direction and the negative s direction.
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361. Patrick 027 at 16:43 PM on 16 December, 2008
     Just to be clear, everything in comments 352 up to here has been focused on (two-dimensional) barotropic Rossby waves. -- For waves that can be described by cos(k*x + l*y - w*t): -- 1. WAVE GEOMETRY (2 dimensions, (x,y)): A WAVE WITH WAVE VECTOR (k,l): phase lines are parallel to: y = -k/l * x because along a phase line: constant = y*l + k*x y*l = - k*x + constant (For y north, x east orientation of x,y: if k and l are both positive or both negative, the phase lines are aligned from northwest to southeast. If they are of opposite sign, the phase lines are aligned from southwest to northeast. If l = 0, the phase lines run from north to south; if k = 0, the phase lines run from west to east.) _______________ wave vector = (k,l) magnitude of wave vector = M = (k^2+l^2)^(1/2) unit vector in that direction = (k,l)/[(k^2+l^2)^(1/2)] = (k,l)/M Direction of wave vector: Angle counterclockwise from positive x direction: A_n cos(A_n) = k/M sin(A_n) = l/M (note: this is the n direction) _______________ unit vector perpendicular to wave vector (to the right) = (l,-k)/[(k^2+l^2)^(1/2)] = (l,-k)/M Direction of (l,-k)/M : Angle counterclockwise from positive x direction: A_s = A_n - pi/2 (radians) cos(A_s) = l/M sin(A_s) = -k/M (note: this is the s direction) ________________ 2. THE DISPERSION RELATION (w as function of spectrum (wave vector)): FOR BAROTROPIC ROSSBY WAVES, with the PV gradient in the positive y direction, and only due to beta, where beta = del(f)/del(y) (Hence, y is north, x is east) (FOR such waves, k is never positive (on a planet with prograde rotation such as most planets, where del(f)/del(y) is never negative) - the wave vector's x component is never positive - the wave vector can be westward, southwestward, northwestward, and in the limit of k=0, it can approach northward or southward - thus, A_n can be between 90 and 270 deg (pi/2 and 3pi/2 radians); A_s can be between 0 and 180 deg (0 and pi radians)). From p.85 Cushman Roisin: w = -beta * R^2 * k / [1 + R^2*(k^2+l^2)] = -beta * k / [1/R^2 + (k^2+l^2)] = -beta * k / [1/R^2 + M^2] = -beta * cos(A_n) * M / [1/R^2 + M^2] where R is the external Rossby radius of deformation, R = (g*H)^(1/2) / f ** NOTICE THAT if 1/R^2 goes to zero (or in the limit of the product (M*R)^2 going to infinity), the dispersion relation above reduces to the proportionality that would occur if the ratio of PV anomaly to RV anomaly were constant: w proportional to -beta * k / M^2, which is equal to -beta * cos(A_n) / M ** w is positive because beta is positive and k is negative. ______________________ PHASE SPEEDS: in the x direction: cx = w/k = -beta / [1/R^2 + (k^2+l^2)] = -beta / [1/R^2 + M^2] (cx is always negative; phase propagation is never eastward.) --- in the y direction: cy = w/l = -beta * (k/l) / [1/R^2 + (k^2+l^2)] = -beta * (k/l) / [1/R^2 + M^2] = -beta / ( tan(A_n) * [1/R^2 + M^2] ) = +beta * tan(A_s) / [1/R^2 + M^2] = (k/l)*cx (cy is positive if l is positive, negative if l is negative; phase propagation is northward for positive l, southward for negative l.) --- in the direction of the wave vector (k,l) (in the n direction): c = w/M = w/[(k^2+l^2)^(1/2)] = -beta * k / ( [1/R^2 + (k^2+l^2)] * (k^2+l^2)^(1/2) ) = -beta * k / [ (k^2+l^2)^(1/2) / R^2 + (k^2+l^2)^(3/2) ] = -beta * k / [ (k^2+l^2)^(1/2) / R^2 + (k^2+l^2)^(3/2) ] = -beta * k / [ M / R^2 + M^3 ] = -beta * k/M / [1/R^2 + M^2] = -beta * cos(A_n) / [1/R^2 + M^2] --- cos(A_n)/c = 1/cx = k/M / (w/M) = k/w cx*cos(A_n) = c sin(A_n)/c = 1/cy = l/M / (w/l) = l/w cy*sin(A_n) = c cy = k/l * cx = cos(A_n)/sin(A_n) * cx ________________________ GROUP VELOCITY x and y components: cgx = del(w)/del(k) = -beta / [1/R^2 + (k^2+l^2)] - -beta * k * 2k / [1/R^2 + (k^2+l^2)]^2 = -beta / [1/R^2 + (k^2+l^2)] + beta * 2k^2 / [1/R^2 + (k^2+l^2)]^2 = -beta / [1/R^2 + (k^2+l^2)] + beta / [1/R^2 + (k^2+l^2)] * 2k^2 / [1/R^2 + (k^2+l^2)] =(-beta / [1/R^2 + (k^2+l^2)] ) * (1 - 2k^2 / [1/R^2 + (k^2+l^2)] ) = w/k * (1 - 2k^2 / [1/R^2 + (k^2+l^2)] ) = w/k - 2*w*k / [1/R^2 + (k^2+l^2)] ) = -beta / [1/R^2 + M^2] + beta * 2k^2 / [1/R^2 + M^2]^2 = +beta / [1/R^2 + M^2] * (2*k^2/[1/R^2 + M^2] - 1) --- cgy = del(w)/del(l) = +beta * k * 2l / [1/R^2 + (k^2+l^2)]^2 = +beta * 2*k*l / [1/R^2 + (k^2+l^2)]^2 = +beta * 2*k*l / [1/R^2 + M^2]^2 _______________ Group velocity vector = [del(w)/del(k), del(w)/del(l)] = [cgx,cgy] Magnitude of group velocity = cg cg = (cgx^2 + cgy^2)^(1/2) = beta * ( 1/[1/R^2 + M^2]^2 * (2*k^2/[1/R^2 + M^2] - 1)^2 + 4*k^2*l^2 / [1/R^2 + M^2]^4 )^(1/2) = beta / [1/R^2 + M^2] * ( 4*k^4/[1/R^2 + M^2]^2 - 4*k^2/[1/R^2 + M^2] + 1 + 4*k^2*l^2 / [1/R^2 + M^2]^2 )^(1/2) = beta *2*k / [1/R^2 + M^2]^2 * ( k^2 - [1/R^2 + M^2] + [1/R^2 + M^2]^2/(4*k^2) + l^2 )^(1/2) = beta *2*k / [1/R^2 + M^2]^2 * ( M^2 - [1/R^2 + M^2] + [1/R^2 + M^2]^2/(4*k^2) )^(1/2) -- IN THE LIMIT of 1/(M*R)^2 = 0, cg = beta *2*k / [1/R^2 + M^2]^2 * [1/R^2 + M^2]/(2*k) = beta / [1/R^2 + M^2] ~= beta / M^2 -- (PS A VECTOR IS EQUAL TO A VECTOR SUM OF all orthogonal components. Any set of orthogonal components can be used.) Component of group velocity parallel to any other vector [a,b]: cg component = cg*cos(angle) = dot product of two vectors = cgx*a + cgy*b cg*cos(angle) = (cgx^2 + cgy^2)^(1/2) * cos(angle) = cgx*a + cgy*b cos(angle) = (cgx*a + cgy*b) / cg ____________ RECAP OF cgx and cgy: cgx = del(w)/del(k) = -beta / [1/R^2 + M^2] + beta * 2*k^2 / [1/R^2 + M^2]^2 cgy = del(w)/del(l) = +beta * 2*k*l / [1/R^2 + M^2]^2 -- Group velocity vector = [ -beta / [1/R^2 + M^2] + beta * 2k^2 / [1/R^2 + M^2]^2 , beta * 2k*l / [1/R^2 + (k^2+l^2)]^2 ] = beta * [ -1 / [1/R^2 + M^2] + 2k^2 / [1/R^2 + M^2]^2 , 2k*l / [1/R^2 + M^2]^2 ] = beta / [1/R^2 + M^2]^2 * [ - [1/R^2 + M^2] + 2*k^2 , 2*k*l ] ________________ Component of Group velocity parallel to wave vector: cgn = (k,l)/M "dot" [cgx,cgy] = beta/M * [ -k / [1/R^2 + M^2] + 2k^3 / [1/R^2 + M^2]^2 + 2k*l^2 / [1/R^2 + M^2]^2 ] = k/M * beta * [ -1 / [1/R^2 + M^2] + 2k^2 / [1/R^2 + M^2]^2 + 2*l^2 / [1/R^2 + M^2]^2 ] = k/M * beta / [1/R^2 + M^2] * [ -1 + 2*k^2 / [1/R^2 + M^2] + 2*l^2 / [1/R^2 + M^2] ] = k/M * beta / [1/R^2 + M^2] * (2*M^2 / [1/R^2 + M^2] - 1) ________________ Component of Group velocity parallel to phase line (to the right of wave vector): cgs = (l,-k)/M "dot" [cgx,cgy] = beta/M * [ -l / [1/R^2 + M^2] + 2*l*k^2 / [1/R^2 + M^2]^2 - 2*l*k^2 / [1/R^2 + M^2]^2 ] = -l/M * beta / [1/R^2 + M^2] *** NOTE: The component of group velocity in the direction of (l,-k) of the wave with wave vector (k,l) is equal to the c of a wave with wave vector (l,-k) (not too surprisingly!). _______________ IN THE LIMIT OF k going to 0, |l| = M, cgs*(|l|/l) = cgx = cx __________________ cgn = 0 when k = 0 or when: 2*M^2 / [1/R^2 + M^2] = 1 2*M^2 = 1/R^2 + M^2 M^2 = 1/R^2 (M*R)^2 = 1 For negative k, cgn is (positive/0/negative) when (M*R)^2 is (greater than/equal to/less than) 1. __________________ group velocity in direction of wave vector relative to phase speed in direction of wave vector cgn - c = = k/M * beta / [1/R^2 + M^2] * (2*M^2 / [1/R^2 + M^2] - 1) + beta * k/M / [1/R^2 + M^2] = k/M * beta / [1/R^2 + M^2] * 2*M^2 / [1/R^2 + M^2] = k/M * 2*M^2 * beta / [1/R^2 + M^2]^2 For waves with negative k, cgn is always less than c. ___________________ IN THE LIMIT OF 1/(M^2*R^2) = 0, cgn ~= k/M * beta / [1/R^2 + M^2] ~= k/M * beta / M^2 cgs ~= -l/k * cgn ~= -l/M * beta / M^2
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362. Philippe Chantreau at 11:53 AM on 24 December, 2008
     To get back on topic (sorry Patrick, you kinda lost me): Arctic sea ice extent growth came to a sudden stop and is now lower than 07. I'd have to verify, but it's likely to make it lower than any previous record for that time of the year. So far NSIDC shows a rather peculiar curve for this year's extent variation: very fast fall growth and now a winter halt.
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363. Patrick 027 at 16:48 PM on 25 December, 2008
     I kinda lost me, too. Didn't mean to get so in depth in a summary... will truly wrap it up later... Tentative utline of what follows: barotropic refraction, reflection, instability, forced planetary waves Why the 1/R^2 stuff? Effect of stratification on otherwise barotropic waves compare baroclinic to barotropic waves The troposphere as a hologram (well, sort'a) --- NAO, AO/NAM, etc...
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364. Quietman at 19:44 PM on 25 December, 2008
     Philippe Interesting, albeit not totally unexpected. NSIDC? Not familiar, can you post a link? Patrick You lost me again as well. Maybe leave out the calculations and proof and go right to the summary?
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365. Philippe Chantreau at 20:59 PM on 25 December, 2008
     Quietman it is hard for me to believe that, after a 360+ posts discussion of Arctic sea ice, you're not familiar with the National Snow and Ice data Center. I linked it several times earlier when WA was appealing to Watts blog and suggesting that the up trend in Antarctic SI was significant whereas the down trend in Arctic was not. A look at these will inform you on that: http://nsidc.org/ http://nsidc.org/data/seaice_index/
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366. Quietman at 13:13 PM on 26 December, 2008
     Philippe Thanks. That is a useful site, better view of the polar caps. The anomally is still exactly where I expected to see it again, I just wanted to confirm it. I have not read all 365 posts. I was busy in the summer, which is why I have not been posting as often also. Unfortunately this is a bad winter and I have been doing a lot of snow removal, salting and sanding since October, so not much time now either.
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367. Philippe Chantreau at 06:05 AM on 27 December, 2008
     From what I read so far, there is some unusual weather in the Arctic that had 2 effects: compacting the ice together (thereby reducing the total extent) and keeping the air much warmer than normal. Neither seemed to have been expected by the people studying the SI. Good luck with the shoveling, hope the winter kills off those bark beetles. We had over 15 inches down in the PDX area, and they're just not used to it here! Of course now it's turned to slush and the roads are nasty. Happy Holidays!
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368. Arkadiusz Semczyszak at 22:26 PM on 31 December, 2008
     Philippe Chantreau - sorry my popularizing (You so, probably think that I’m not "scientist"?) publications are in a ”beautiful” (very difficult too), polish language - on-line, for example only one: http://192.168.0.11/download/technologia/efekt_szklarni.pdf. I propose You this page about “ice”: http://noconsensus.files.wordpress.com/2008/12/global-sea-ice-area-variation-bootstrap-algorithm1.jpg?w=667&h=455 (- do not have decreasing sea ice at the last 30 years?) and http://www.unep.org/geo/geo_ice/images/full/6a_antarcticamassbal.png - for a complete image about “The global ice story”. …and may dear “Interlocutors” (hi, hi, hi…) Happy New Years everybody!!!
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369. Philippe Chantreau at 07:58 AM on 1 January, 2009
     Let me clarify one thing: in my definition, a scientist is someone who is an expert in a field (usually that implies an advanced degree in that field or a closely related one), does research (i.e. publishes papers in peer-reviewed science journals) and whose work is of significant interest to others doing research in that field (i.e. cited, used in other publications). If that's you, then you're a scientist. If not, then you're not one by that definition, and I like that definition, that's the one I use. You said that you were doing applied science. That's vague. A pilot does applied physics and engineering, that does not make him a physicist or aeronautical engineer. The noconsensus graph is not referenced, I have no idea where it comes from. Is area really more important than extent and volume (this latter being certainly the most important)? The Antarctic mass balance graph has no legend and it really needs one; still, it's unclear at first glance whether there is a net gain or loss, I see sizeable areas in the graph with the rectangles showing considerable negative gain (loss?). At my level, I find NSIDC and Cryosphere Today much more useful: http://nsidc.org/sotc/sea_ice.html http://www.nsidc.org/data/seaice_index/ This graph shows a statistically significant decline in global sea ice: http://arctic.atmos.uiuc.edu/cryosphere/IMAGES/global.daily.ice.area.withtrend.jpg This does not show much of a TREND in Antarctic sea ice anomaly: http://arctic.atmos.uiuc.edu/cryosphere/IMAGES/current.anom.south.jpg Unlike the TREND shown here for Arctic sea ice: http://arctic.atmos.uiuc.edu/cryosphere/IMAGES/current.anom.jpg Quote from NSIDC, from the Arctic and Antarctic standardized anomalies and trends Jan 1979 Dec 2007 graph: "The Antarctic ice extent increases were smaller in magnitude than the Arctic decreases, and some regions of the Antarctic experienced strong declining trends in sea ice extent." The noise in the Antarctic is larger because of a much stronger seasonal variation. "
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370. Patrick 027 at 16:22 PM on 6 January, 2009
     1. (as if my numbering scheme makes any sense!) Why the 1/R^2 stuff? The dispersion relation (frequency as a function of wavevector) from Cushman-Roisin, stated in comment 361, was based on a basic state vorticity gradient equal to beta in the y direction ( beta = del(f)/del(y) ), due entirely to variation in f, with basic state RV being constant and zero; there is no basic state wind - Oh, and there is no internal stratification (zero static stability; constant potential temperature (or potential density in the ocean)). The same dispersion relation is found (in Cushman-Roisin - p.89) to apply to a situation with no basic state wind, constant f, but varying underlying topography, where beta (which is equal to the basic state barotropic PV times the depth of the fluid) is replaced with the depth of the fluid times the basic state PV gradient which is due to varying fluid depth. In other words, the propagation of barotropic Rossby waves is the same for the same barotropic PV gradient, (where barotropic PV is proportional to AV/H, where H is the depth of the fluid; this is most obviously applicable to a nearly incompressible fluid with a top and bottom such as the ocean, but I think it can be made to apply to the atmosphere if H is taken to be proportional to surface pressure ** - the important thing is that H be proportionate to mass per unit area within a fluid layer), whether that gradient is due to variation in f or variation in fluid depth - I presume the relationship would apply to any such barotropic PV gradient due to any combination of beta and variable H. A key thing about Rossby waves is that they are quasigeostrophic or at least nearly so. The balance between wind and mass fields is approximated as geostrophic and while it is kept in mind that there is ageostrophic and vertical motion required to keep imbalances from growing indefinitely, it is assumed that the imbalance is never sizable - most of the wind field can be approximated by the geostrophic wind. This is an appropriate approximation for this purpose for much of the atmosphere except at low latitudes and except for small spatial scales and near the surface (Hence, near the equator, the gap in the spectrum of atmospheric waves between quasigeostrophic Rossby waves, and those fundamentally ageostrophic inertio-gravity and Kelvin waves and inertial oscillations, dissappears - and so there are such things as equatorial Rossby-gravity waves). But for Rossby waves to remain in near geostrophic balance, the RV changes induced by advection of PV must be balanced by changes in the horizontal pressure gradient (which in a barotropic fluid, results from variations in H due just to variations in the 'top surface'). With no preexisting pressure variation, there must be divergent or convergent motions, which bring the RV due to conserved PV and the geostrophic RV closer together. Which one budges more is wavelength dependent. For large wavelengths (which have small wave vectors), the same geostrophic RV requires larger winds and also that the pressure gradient extend over longer distances, so that the variation in pressure should scale with geostrophic RV * square of wavelength. The divergence required to accomplish this reduces the resulting RV variation per unit PV variation, thus slowing the phase speeds and reducing the frequency relative to what they would be if AV were conserved. The shortest wavelengths will have much less pressure variation over a wavelength, so that the same amplitude of PV wave produces a larger RV wave; in the limit of shortest wavelengths, AV is conserved. Of course, at that shortest wavelengths, the geostrophic approximation breaks down. IF, however, there is a basic state wind (necessary but not sufficient for some basic state RV gradient), then assuming it is nearly geostrophic, there is a basic state pressure variation. Advection of pressure variation along with PV variation can/will alter the divergence necessary to maintain near geostrophic balance, and thus the dispersion relation, phase speed, and group velocity patterns may be different. PS could the RV wave ever be 180 degrees out of phase with the PV wave? In that case (if it is possible - I'm not sure - maybe if the RV gradient was in the opposite direction as the beta and topographically-caused PV gradients, and there were an easterly basic state wind ???), waves would propagate in the opposite directions as previously described. But even then, some general concepts described in previous comments would still apply somehow. _________ 2. What if there is stratification? - nonzero static stability (which will be designated here as S which is equal to the negative vertical derivative of potential temperature with respect to pressure (or potential density with respect to ... some measure of depth in the ocean): S = - del(q)/del(p). Well, then there is not a wave which doesn't vary at all in height; but a nearly barotropic wave can exist. Such a wave is modified such that (at least setting aside what a basic state wind would imply) amplitude is larger near the surface for topographic waves and smaller near the surface for waves due to variation in f, as decribed in comment 322 above. Pressure systems associated with Rossby waves that are supported by beta would be cold-core lows and warm-core highs; whereas pressure systems associated with topographic Rossby waves would be warm-core lows and cold-core highs. There are also fully baroclinic modes. Cushman-Roision derived a dispersion relation (next comment) for horizontally propagating Rossby waves that reverse phase one or more times in the vertical; a vertical cross section would appear as a checkerboard pattern. For relatively weak waves, these baroclinic waves, as with the barotropic ones, can be mathematically and qualitatively analyzed as the result of linear superpositions of other waves or an infinite number of point anomalies or finite number of anomalies of finite size, etc.; hence, the checkerboard pattern could be thought of as a wave which propagates in the horizontal but is a standing wave in the vertical direction, resulting from two sets of baroclinic waves that propagate in the same direction horizontally and in opposite directions vertically, and turn into each other by reflection from top and bottom boundaries (I think in terms of the RV wave, the reflections are in phase with the incident waves, so that (some of the) RV maxima and minima, but not the vertical nodes, occur at the top or bottom - at least in the case of a fluid layer with definite top and bottom with no overlying or underlying fluids of comparable density ?).
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371. Arkadiusz Semczyszak at 19:39 PM on 6 January, 2009
     Philippe: First: I only cites publications pure scientists, not my working. Both: By ten years I was research the correlations: meteorological conditions - aphids - Entomophthoraceae; together with the scientists from Yakima and federal department for Agriculture in USA)., next a twelve years as adviser for agro-meteorology - if it for You too small - sorry… …but and I don’t understand, why You don’t cites, in your post, this figure: current_anom_south0325.jpg? (f. e. from page: http://www.smalldeadanimals.com/archives/008350.htm or http://icecap.us/images/uploads/current_anom_south0325.jpg.) It’s very interesting yet in “this Theme” ! But I think probably not too inconvenient for You, certainly (that for IPCC, I understand, hi, hi, hi…) ?!
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372. Arkadiusz Semczyszak at 20:04 PM on 6 January, 2009
     It signify walk me about comment - your cites figure http://arctic.atmos.uiuc.edu/cryosphere/IMAGES/current.anom.south.jpg - the newest than current_anom_south0325.jpg - What You think about it - It’s La NIna - ENSO, AMO effects ?
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373. Arkadiusz Semczyszak at 21:31 PM on 6 January, 2009
     The polish professor A. Marosz has analyzed depth Antarctic sea ice cover. Sorry I find only polish language ( http://ocean.am.gdynia.pl/wydaw/Marsz1_aa2007.pdf), however, I’m proper seeing - even if only figures or tables, especially Fig. 1. (by polish - Rys. 1.) The 1 and 2 column of table - it’s average, 3 columns - it’s standard deviations . I will cite one - finale conclusion, in this paper: The Changes within last 50 yrs was only natural (influence AGW don’t statistic important), if walks about the area shelf and sea ice…
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374. Philippe Chantreau at 07:39 AM on 7 January, 2009
     Arkadiusz, I don't understand what you mean by too small. I told you what I consider a scientist, it is far from being an unusual definition; I'll leave it to you to determine whether or not you meet the criteria. I don' care that much myself, don't take it personally Your link to the deadanimals blog leads nowhere. I had never heard of that site and, at first glance, it looks like a very poor source of scientific information. Why would I get information on polar ice from political blog? The reason I use NSIDC and Cryosphere today is very simple: they have teams who study the ice and compile data all the time. That's what they do. I don't understand why it seems to bother you that I don't go to blogs to get that kind of information. I note, on the other hand, that most of your links are from blogs, even when they are actually leading to the real stuff, like cryosphere today. You ask about the southern hemisphere SI anomaly. There is hardly a trend there, it barely makes it out of the noise, what exactly are you asking? Depending what error bars you use, you could find no trend at all. If I was to show a graph like that going the "other direction" (provided there is one), I can only imagine how summarily "skeptics" would dismiss it. Well, I'm affording myself that same luxury, usually reserved for so-called "skeptics." Climate change has not yet abolished the Southern Annular Mode, as far as I know. Ther is a very small upward trend in Southern polar sea ice. I linked this paper earlier: Author(s): Zhang JL Source: JOURNAL OF CLIMATE Volume: 20 Issue: 11 Pages: 2515-2529 Published: JUN 1 2007 Times Cited: 1 References: 34 Abstract: "Estimates of sea ice extent based on satellite observations show an increasing Antarctic sea ice cover from 1979 to 2004 even though in situ observations show a prevailing warming trend in both the atmosphere and the ocean. This riddle is explored here using a global multicategory thickness and enthalpy distribution sea ice model coupled to an ocean model. Forced by the NCEP-NCAR reanalysis data, the model simulates an increase of 0.20 x 10(12) m(3) yr(-1) (1.0% yr(-1)) in total Antarctic sea ice volume and 0.084 x 10(12) m(2) yr(-1) (0.6% yr(-1)) in sea ice extent from 1979 to 2004 when the satellite observations show an increase of 0.027 x 10(12) m(2) yr(-1) (0.2% yr(-1)) in sea ice extent during the same period. The model shows that an increase in surface air temperature and downward longwave radiation results in an increase in the upper-ocean temperature and a decrease in sea ice growth, leading to a decrease in salt rejection from ice, in the upper-ocean salinity, and in the upper-ocean density. The reduced salt rejection and upper-ocean density and the enhanced thermohaline stratification tend to suppress convective overturning, leading to a decrease in the upward ocean heat transport and the ocean heat flux available to melt sea ice. The ice melting from ocean heat flux decreases faster than the ice growth does in the weakly stratified Southern Ocean, leading to an increase in the net ice production and hence an increase in ice mass. This mechanism is the main reason why the Antarctic sea ice has increased in spite of warming conditions both above and below during the period 1979-2004 and the extended period 1948-2004."
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375. Philippe Chantreau at 12:56 PM on 7 January, 2009
     Sorry, I don't read Polish at all. What journal was this last document published in?
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376. Patrick 027 at 16:16 PM on 8 January, 2009
     ..."barotropic PV is proportional to AV/H, where H is the depth of the fluid; this is most obviously applicable to a nearly incompressible fluid with a top and bottom such as the ocean, but I think it can be made to apply to the atmosphere if H is taken to be proportional to surface pressure ** - the important thing is that H be proportionate to mass per unit area within a fluid layer"... That last part is indeed the important thing when considering how PV varies with changes in H. However, my impression is that H must be an actual vertical scale to be correctly used in the Rossby Radius of Deformation R, where: external R = sqrt(g*H) / f internal R = N*H / f or is proportional to N*H / f ------------------- "For relatively weak waves, these baroclinic waves, as with the barotropic ones, can be mathematically and qualitatively analyzed as the result of linear superpositions of other waves or an infinite number of point anomalies or finite number of anomalies of finite size, etc." And so one might consider what the vertical cross section of the wind field would be for a given IPV anomaly. (Note that IPV/g = S * AV; S is inversely proportional to mass per unit area in between two isentropic surfaces of a set difference in q; hence, IPV is like a barotropic PV defined for incremental isentropic layers of air (or incremental layers of constant potential density within the ocean - in that case it wouldn't be called isentropic PV, but it would serve the same role in fluid dynamics). ) In a horizontal plane, the wind field of an RV anomaly isolated in both dimensions decreases in strength away from the RV anomaly, being proportional to 1/distance, and is directed in opposite directions on opposite sides of the anomaly; within the anomaly the wind field increases in strength out from the center. An IPV anomaly, when the atmosphere is nearly in geostrophic balance (or else a gradient wind balance)with it, will have induced a column of RV anomaly that extends above and below it. In the horizontal planes, the wind field of the RV anomaly is as described above. In the vertical direction, the RV anomaly and it's wind field generally will decay in strength away from the IPV anomaly - exponentially or roughly so if certain conditions occur (such as some parameters being constant in height or varying in just the right way, some approximations, and also, that the anomaly is relatively weak). Given such conditions, the rate of this decay (inversely proportional to the height scale in pressure coordinates, Hp), is, in pressure coordinates, proportional to the square root of S. It is less for IPV anomalies with larger horizontal extents/wavelengths (the length scale L) and for larger f and larger basic state AV. More specifically, Hp is proportional to L * sqrt(f*AV)/sqrt(S). The Height scale in isentropic coordinates (Hq if it comes up here again) varies the same way except that sqrt(S) would go in the numerator; this is simply because of the geometry of variation in p (pressure) relative to q (potential temperature) implied by S. PS I am using 'q' in place of the greek letter 'theta', which is q in a symbol font; In textbooks you will see q used for other quantities such as quasigeostrophic potential vorticity given in units of vorticity - watch out! PS In the above, L is representative of length scales in both horizontal dimensions - to be more precise, I think it could be given in terms of two orthogonal length scales X and Y as L = 1/(1/X + 1/Y) ?? ...
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377. Quietman at 08:47 AM on 9 January, 2009
     Arkadiusz Semczyszak Your link: http://arctic.atmos.uiuc.edu/cryosphere/IMAGES/current.anom.south.jpg Shows why sea level rise isn't a problem. From that chart it looks like what melted in the arctic has refrozen in the antarctic. I would think it relates to the 2007-08 La Nina (ENSO). ps Phillipes definition of a scientist is not correct, just his opinion. Many scientists do not publish papers because they work in the private sector and their research is the property of their employer. His definition is just academic snobbery. We have a saying in my country, "those than can, do while those that can't, teach. Sorry Phillipe, you son't have to speak polish to insult someone, english does you just fine. In Europe and Asia, the term "professor" is used to indicate a scientist rather than a teacher. I would think that you would know this since you said that you spent time in Europe.
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378. Quietman at 08:51 AM on 9 January, 2009
     Patrick Way too much detail. Take a break and look at the links from Arkadiusz Semczyszak and give us your opinion (in brief please).
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379. Quietman at 08:56 AM on 9 January, 2009
     ps "Times Cited: 1 References: 34 " It only means that 34 authors agreed with the argument, nothing else. It does not lend credulity.
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380. Patrick 027 at 12:29 PM on 9 January, 2009
     Actually, L ~= 1/sqrt(1/X^2 + 1/Y^2) because 1/L^2 ~= 1/X^2 + 1/Y^2 PS in case anyone was confused by this, some of what I've written uses Microsoft Excel language: sqrt(x) = square root of x, x^3 = cube of x, etc... The relationship between L and Hp can be simplified to Hp is proportional to L * f/sqrt(S) if AV is approximated by f - which is good first approximation for much large scale motion of the atmosphere. N is proportional to sqrt(S) (for a given p and q), so if AV ~= f, then Hp is proportional to L * f/N which means that L is proportional to the internal Rossby radius of deformation for a height scale Hp. Perhaps also, then, the Rossby radius of deformation might be more accurately given by R is proportional to H*N / sqrt(f*AV) (For a given p and q, a small relative change in p is roughly proportional to a change in geometric height z). And if a gradient wind balance is used, then f may be replaced in the above with f_loc (see last part of comment 349, or Bluestein p.190). (Around a center of cyclonic rotation, the magnitudes of both f_loc and AV will be greater than otherwise, which suggests that for a given height scale, the Rossby radius of deformation is smaller in cyclones than in anticyclones; or for a given length scale of an IPV anomaly, the vertical scale of the induced RV and wind anomalies will be larger in cyclones than in anticyclones. Latent heating during ascent mitigates the dynamic effect of S, so the change in H/L or H/R for cyclones vs anticyclones should be enhanced for cyclones with precipitation.) ---------- (PS the reason for decreasing induced RV anomaly with vertical distance from an IPV anomaly, and the dependence of that relatiohship on S (or N), AV and f, and L, is that vertical stretching or contraction, which occurs with horizontal convergence or divergence, respectively, is necessary (for isentropic vorticity, without latent or radiative heating/cooling and without friction or mixing) to change RV, and horizontal variation in vertical motion in the presence of a nonzero S or N results in horizontal temperature variations that allow a change in RV over vertical distance to be in geostrophic balance or gradient wind balance. See also comments 313 and 319 above. ---------- Notice that the total vertical extent of the whole fluid (atmosphere or ocean) limits how much convergence and divergence of other layers of the atmosphere can adjust to an IPV anomaly at some level. Hence, ** Less total fluid depth might increase the RV anomaly at all levels that result from a given IPV anomaly???) And if an anomaly occurs at an upper or lower boundary, my impression and understanding is that the RV field is doubled in strength (but has half the volume)... Which makes me think that vertical variations such as an increase in S at some level will partially reflect the RV field induced by an IPV anomaly; and that reflection will be in phase with the incident RV field... Would a decrease in S result in a reflected RV field that is out of phase? These are things I have yet to figure out.) ---------- Variations in basic state properties can/will distort the RV and wind fields from the above description (see last "PS" section) (PS for the atmosphere in particular, Holton p.412-419 finds solutions for vertically propagating waves of various kinds (including Rossby (planetary in particular)) which increase in amplitude with height, in proportion to 1/sqrt(basic state density), which makes me wonder if the RV field of an atmospheric IPV anomaly will tend to be stronger above the IPV anomaly than below it?), but they should generally be qualitatively similar. ---------------- PS: Concerning the value of RV at the IPV anomaly (where 'subscript' 0 refers to basic state values and a ' indicates anomaly values): IPV'/g = RV'*S + AV0*S' = RV'*(S0+S') + AV0*S' For a given IPV', RV' may be roughly proportional to IPV'/S if AV0 and/or S' are small. However, under other circumstances, RV' may be between being proportional to IPV'/S and being proportional to Q_*IPV'/[S0^(3/2) * L * sqrt(f*AV^2/AV0)], where Q_ is the vertical thickness of the IPV anomaly itself, in terms of q coordinates. In terms of p coordinates, the thickness, P0, is equal to Q0/S. Of course, S changes at the IPV anomaly, and changes in the opposite way above and below it (although if Hp is much larger than P0 or Hq is much larger than Q0, the S' above and below the anomaly, which decays with vertical distance away from the anomaly as does RV', will be much less than the S' that occurs within the IPV anomaly). One simplication to the math (which was necessary even just to get some of the above relationships) is to assume S' is much smaller than S0, so that S is nearly equal to S0; such is the case with weak anomalies... --------- Anyway...
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381. Philippe Chantreau at 14:50 PM on 9 January, 2009
     Actually it means that the paper has been cited 34 times in other articles, I believe. I also believe that it's pretty darn good and indicates that the paper is very relevant to other's research. Not that they "believe" in it. What a strange way to look at it. I do fine in English indeed, although it is my second language. Credulity is something that many would like to lend, and strangely enough, many seem willing to borrow... I like my definition, it is shared by most scientists, and I'll keep using it, that's entirely my prerogative, just like you think it's yours to impart disproportionate weight to non published ideas. I don't know what the heck you're trying to say with the mumbo-jumbo on teaching, professors and what not. I'm still curious to know what journal was Marosz pdf published in. Or was it an opinion piece?
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382. Patrick 027 at 15:43 PM on 9 January, 2009
     Quietman - Okay, there are scientists who don't publish in publically-available forums. (How many of those would be working professionally on climatology? - well I suppose the Pentagon...?) PS I am not actually a scientist (yet). (379)--"It only means that 34 authors agreed with the argument, nothing else." Does anything mean anything else? Taken all by itself I suppose that may be about it, but there's a context there... (377) - "Shows why sea level rise isn't a problem. From that chart it looks like what melted in the arctic has refrozen in the antarctic. I would think it relates to the 2007-08 La Nina (ENSO)." I don't know quite in what way ENSO is connected to that, but sea ice melt and growth has little direct impact on sea level (the little bit it would have comes from the ocean not being fresh water; otherwise it would be none at all). Sea ice has an indirect effect by holding land glacier flow into the sea back a bit (when in the form of ice shelves), and obviously will have other indirect effects via climate (albedo, local surface characteristics, affects on wind/water momentum transfers, ecology...). What affects sea level is melting and/or transfer of ice supported by land/rock to the ocean, and the density of the water (affected by temperature and salinity), and regionally, variations in those and in in the wind. And in the longer term, isostatic adjustments of the crust, and in the much longer term, plate tectonics/continental drift/mantle convection... (and in the much longer term, the chemistry and dynamics of the pre solar nebula ! :) ) And also, the global trend is not zero, from what I've been hearing... (377)- ""those than can, do while those that can't, teach." What about those that can teach? :) PS much of my motivation here is that teaching is a great way to learn; having a potential audience (I'm pretending at least some people are reading my comments :) ) is a great motivation to prepare. (378) - "Patrick Way too much detail. Take a break and look at the links from Arkadiusz Semczyszak and give us your opinion (in brief please)." I think globally sea ice changes have been significant and are worrisome (We've already had a taste of some political ramifications (Putin,etc.)). The explanation quoted by Philippe (374) about Antarctic sea ice is quite interesting - and sounds familiar - did I see it earlier somewhere? - well he said himself that he mentioned it or referenced it earlier... Is some of it related to AO/NAM (and in Antarctica, SAM)? Well, I suppose it could be - some probably is (although without knowing more specifics, there is the possibility that the portion is a negative fraction - ie that an opposite trend would be attributed to AO/NAM - which would mean everything else has to account for over 100 % - just as everything besides aerosols has to account for over 100% of observed warming... (PS I'm not saying - about NAM/SAM - that I think that this is the case; I mention it just to cover the bases). But even if that is, some of NAM and SAM trends are not 'natural' - in that they are anthropogenically-forced. What fraction? I really don't know. I do know ozone depletion would cause an increase in SAM in particular and may have some contribution to NAM (and increased CO2,CH4,etc. could exacerbate polar ozone depletion). I also know that at least some model(s?) have reproduced some increase in NAM as a result of greenhouse gas increases... And I'm still not sure I understand the causal link, but I have found a couple papers (suggested at RealClimate, thanks!) and am part way through the second. The problem is there is this other paper I also found which argued that the proposed mechanism wouldn't work ... BUT I can think of some other mechanisms... And then there's the whole tidal-forcing concept. It's intriguing but I'm skeptical. In case I don't get to it later: It seems more likely, based on the argument put forward, that stronger tides would cause more cooling than weaker tides would cause warming. Also, I saw no mention of the changes in the eccentricity of the lunar orbit, so I wonder how accurate the judgement of periods of several strong tides or lack thereof would be... The idea that variations are big enough over such timescales is hard for me to see - but here are some ideas: changes in area of exposed ocean at high latitudes due to changes in tidal currents that drive ice and icebergs around each other or islands or sea floor bumbs, and affecting ocean mixing via that... AND, driving tidal currents through hydrothermal vents, cooling the vents, thus increasing geothermal heat transfer back into the vents and the ocean (but notice how localized and small an effect that would be)... Other stuff, in brief: CO2 doesn't just go up and down a lot in the bulk of the atmosphere. (You'll find some papers about changes in ~100(?) ppm over hours - well of course, that's under the canopy of a forest, - or maybe in city streets with variations in traffic?? - The point being it's a small volume of air and not climatologically significant, at least not outside of microclimates (and then, only indirectly via effects on plants, etc., I would guess). My understanding is that outside of human activity (or maybe including it), it would take a catastrophic phenomenon to cause CO2 to change as much as it has as fast as it has - at least over the last few decades (even calculating how fast it would appear to have happenned if found in the ice core record (which can smooth out some things) at some later time, it still dwarfs, in terms of sustained rate of change, anything in at least the last ~20,000 years - that includes the end of the last ice age - see IPCC AR4 WGI Ch.6) "Take a break" I did! :)
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383. Patrick 027 at 16:08 PM on 9 January, 2009
     "Which makes me think that vertical variations such as an increase in S at some level will partially reflect the RV field induced by an IPV anomaly; and that reflection will be in phase with the incident RV field..." By analogy - the ocean, sandwiched between the atmosphere and the crust, is analogous to a fluid layer of finite static stability sandwiched between other fluid layers of near-infinite stability. ______________ ...So, consider, with a basic state IPV gradient in the y direction: 1. the pattern of IPV advection around an IPV anomaly as seen in the x,y plane (which would be qualitatively similar to barotropic PV advection around a barotropic PV anomaly). 2. the pattern of IPV advection about an IPV anomaly as seen in cross section in the x,z plane (or x,p plane or x,q plane - whichever vertical coordinate you want (there are more: log-pressure coordinates, sigma coordinates)). The mathematical details are different and will be altered by variations in basic state, but qualitatively there are essential similarities. Which implies that, setting aside variations of anomaly IPV in the y direction, the pattern of IPV anomaly in x,z will propagate in a way similar to the propagation of IPV or barotropic PV patterns in x,y, setting aside variations in z. SO, now I understand how and in which direction Rossby waves of a given tilt in x,z will propagate vertically.
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384. Philippe Chantreau at 16:20 PM on 9 January, 2009
     Thanks for the break, Patrick. Rossby wave propagation and transformations are beyond me but I can sense the aeshetic of it and I think I understand why you're into it :-)
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385. Quietman at 10:20 AM on 10 January, 2009
     Patrick Thank you. Regardless of definition I consider you a scientist already because of your methods and studies, I am sure a Ph.D. will be there eventually. I appreciate your honest and thought out answers.
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386. Quietman at 10:25 AM on 10 January, 2009
     Phillipe By the 34 I refer to the fact that it is a one sided view. How many did not use this as a reference because they disagree? So the number, in itself is both useless and redundant (any only seems to occur in a refuted subject). I have never seen a "number referred" in a paper in ANY other field. So why is it there? To ATTEMPT to convince, and nothing more.
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387. Philippe Chantreau at 10:40 AM on 10 January, 2009
     Nonsense. Look for a paper in Google scholar and you'll see how many references were made to that paper. Find a paper on GRL and they have a tab dedicated only to that purpose ("cited in" tab) that even tells you what papers it was referred in and gives a link to that paper if available.
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388. Patrick 027 at 12:24 PM on 10 January, 2009
     From 380: " And if an anomaly occurs at an upper or lower boundary, my impression and understanding is that the RV field is doubled in strength (but has half the volume)" Because, if the conditions are such that the RV distribution is vertically symmetric about an IPV anomaly, then there is no vertical displacement (in terms of q and p relative to each other) on the quasi-horizontal plane (the isentrope?) that cuts through the IPV anomaly. One can remove the top or bottom half of the IPV anomaly and the RV field and put the remainder against a boundary. Since each half of the IPV anomaly should have contributed something to the RV field at each level, this suggests that the remaining RV field in the half volume is near twice the strength of that which would be produced from the remaining IPV anomaly if not next to such a boundary; hence, the RV field that would have been produced on the other side of the boundary has been added instead to the first side... etc.. I think ... ________________________________ As with gravity waves (and, as I recall from Holton Ch. 12, equatorial Rossby-gravity waves and Kelvin waves), the component of vertical group velocity along phase lines in the vertical plane is upward if the phase tilts with increasing height towards the direction of horizontal phase propagation (For Rossby waves, toward the west if the IPV gradient is to the north) and downward if oppositely tilted. In general, the vertical component of group velocity (which includes a contribution from the component of group velocity perpendicular to phase planes) for all these waves is oppositely directed from the vertical phase propagation. _____________________ Rossby waves have been described by their propagation through the air. Propagation relative to the surface requires adding the basic state wind. For a basic state PV gradient to the north (which is the default condition since the beta effect tends to dominate over much or most of the atmosphere), for a given wind speed to the east, there is a portion of the spectrum (in particular, for barotropic waves with phase lines aligned north to south, one particular wavelength) for which the waves are stationary relative to the surface. Disturbances may force Rossby waves. In particular, some, such as mountain ranges, persistent areas of deep convection over SST anomalies, and land-ocean thermal contrasts (?), form stationary patterns (or nearly stationary over shorter time periods). The large scale wind can be altered by these things into wavy patterns. These waves would tend to propagate; however, the forcing propagates through the air according to the basic state wind; thus, some portion of the spectrum which would freely propagate in the same way can be resonantly excited by such forcing (at least, if those wavelengths are present in the forcing pattern). From Holton p.220 - 222 (considering a simple but useful illustrative model), with no damping (friction, etc.), the amplitude of the resonant wave becomes infinite over time; shorter or longer wavelengths have troughs (vorticity maxima) and ridges (vorticity minima - in usual wave terminology, that would be the trough, but it is called a ridge because of how it appears in streamline patterns in the Northen hemisphere westerlies) are in phase with topographic maxima and minima, with the matchup dedending on whether the wavelengths are shorter or longer than the resonant wavelength. Adding boundary layer drag to damp the wave amplitudes, an infinite wave amplitude is avoided, but there is still a resonant response centered at the same wavelength, and the ridges are 1/4 wavelength upstream from mountains while the troughs are 1/4 wavelength downstream. The very simple model (Charney-Eliassen) does a good job reproducing the observed pattern of the Northern Hemisphere midlatitudes at the 500 mb level (p.222). These quasistationary waves are nearly barotropic, but not quite - there is some westward tilt with height, suggesting that wave energy is propagating upward, with the upward group velocity. This makes sense since, at least for the topographic forcing, the source is at the surface. (Other sources within the troposphere would still contribute to upward group velocity into the stratosphere). __________________________ Variations in basic state conditions (stability, vorticity, gradients of these things) will cause spatial variations in Rossby wave propagation, and thus cause refraction and reflection. And from what I've read, absorption and over-reflection (which I suspect is analogous to the stimulated emission of radiation) can occur. (I think it could also be said that there is emission - as in when a disturbance is introduced into the atmosphere from a non-Rossby wave source Such sources would include baroclinic and barotropic instability). One could view that in two steps: Changing the distribution of RV for a given IPV pattern, and changing the IPV advection pattern for a given RV pattern. Variations in the basic state wind can also distort the shape of IPV anomalies and thus change the wave vector and wavelengths of Rossby waves and therefore also affect propagation. For example, a wave packet with some group velocity might encounter horizontal or vertical shearing, which reorients the phase tilts, potentially reversing the group velocity.
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389. Quietman at 15:46 PM on 10 January, 2009
     Google scholar ?
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390. Patrick 027 at 16:28 PM on 10 January, 2009
     Nonlinearities: In addition to finding barotropic and baroclinic Rossby waves in a stratified fluid (p.214-219 in Cushman Roisin), Cushman-Roisin also finds mathematical solutions in the form of propagating vortices (p.219-223). Typically, one would have to find multiple wave packets to linearly superimpose to create some isolated anomaly, and each component may have it's own group velocity, so the anomaly will spread out in time (unless it just happens to on track toward some maximum compactness, in which case it will spread out afterward). (PS mathematically, this is actually closely related to the Heisenberg Uncertainty Principle: to construct the wave form for an electron with some location, one must put together many sinusoidal waves. A Gaussian distribution for location can be constructed from the linear superposition of an infinite number of sinusoidal waves, each weighted by a function of it's wave number, which is proportional to momentum; for the total to have a Gaussian shape, the weighting is a Gaussian shape over wave numbers. It turns out that the more tightly confined (smaller standard deviation) the Gaussian distribution in location, the less tightly confined (larger standard deviation) the distribution of wave numbers, and vice versa. Hence, one cannot know both the position and momentum of an electron to arbitrarily great precision.) ... And one cannot expect just any vorticity anomaly pattern to retain it's form over time while propagating. But due to nonlinearities, some forms can. I think, for waves in general, such forms might be called solitons (?). Holton, p.349, suggests some blocking patterns (somewhat persistent quasi-stationary high amplitude waves in the flow pattern) may be such "solitary waves" which maintain amplitude in the face of dispersion by nonlinear advection. (Other blocking can be caused by fluxes by transient waves). The vortices described by Cushman-Roisin (p.221-223) are such phenomena. They have rather interesting behavior in that, while the longest wavelengths of Rossby waves with PV gradient to the north will propagate westward with the largest speed for Rossby waves, and the shorest wavelengths will barely propagate, the smallest vortices propagate to the east and the largest propagate to the west, but the closer each is to the cutoff between them, which is something like the Rossby radius of deformation, the faster the propagation... __________________ Wave-mean interaction I: Holton Ch.10, 349-351, describes a very simple model that illustrates a form of low-frequency variability with topographically forced Rossby waves (see comment 388). This simple model takes external climate forcing into account as a basic state equilibrium zonal wind speed. Rossby waves are excited by flow over topography. Vorticity fluxes by Rossby waves **(more on that in a little bit) and the drag force (form drag) due to flow over topography (via pressure variation from Rossby waves) can act to change the basic state wind, which is otherwise tending to approach the equilibrium basic state wind (which physically could be determined by the distribution of heating and cooling). Under some conditions, three equilbrium states can be found; one is unstable and the other two are stable. Of course reality is much more complex but the chaotic switching in between regimes could be understood as a consequence of something somewhat similar to multiple equilbria. ______________ Instabilities: (PS do you like how fast this is going now?) ... has to wait till tomorrow.
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391. Patrick 027 at 16:39 PM on 10 January, 2009
     Seemingly odd analogies: 388 - "are in phase with topographic maxima and minima, with the matchup dedending on whether the wavelengths are shorter or longer than the resonant wavelength. " Which is actually analogous to what happens when water flows over an obstruction at speeds faster or slower than the gravity wave speed in the water (I think the cutoff is near a Froude number of 1; when near 1 it is possible to have a hydraulic jump downstream of the obstruction.) (And this is also analogous to flow through a tube with a narrow section, with a Mach number either less than or greater than 1 - near 1, and it is possible to start subsonic and end up supersonic, etc...)
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392. Philippe Chantreau at 20:37 PM on 10 January, 2009
     Yes, Google Scholar. A tool to find science papers, authors etc. I don't understand your hang up about this. When you research something, you start with a review of the litterature. When you find an article, you obviously look at the references. But you also look at the cites. Why? Because if the article was cited in another paper, chances are that paper can be interesting for your research. It is a fairly universal thing. Just look up papers, you will see the cites.
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393. Quietman at 07:24 AM on 11 January, 2009
     In the REFERENCES section of the paper, all citations used for that paper are listed. What else is actually required? This is always how it was done. Google is an algorthym, it may or may not function as you might think.
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394. Philippe Chantreau at 12:32 PM on 11 January, 2009
     You're still not getting it. This may be how it was done years and years ago. I don't think that anyone ever came up saying "let's require authors to publish cites too." It happened to become the norm because it is so useful to those doing research and computers are so well suited for that kind of stuff. How long as it been since you've done any kind of research, even a superficial cursory search online? OK, look: this is a normal thing, everybody in science does it. Look this up: http://www.pnas.org/content/100/18/10393.abstract Not only that article has a complete list of citing aricles, but even a link for every one of them. The PNAS page even gives you an option of being alerted if the article is cited. Elsevier does the same. All science journals online give you these kind of options. You just never noticed because you never paid attention. Get out there, do some searches and use the tools. Geez...
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395. Patrick 027 at 12:55 PM on 11 January, 2009
     (388 clarification): The similarity between all three is that there is a switch, in going from flow slower than free wave phase speeds, to faster flow, in the positioning of Rossby wave rigdes and troughs, maxima and minima in water level (corresponding to the same in potential energy and the reverse in kinetic energy), and maxima and minima in pressure (corresponding again to potential energy and the reverse in kinetic energy). The extra similarity between the water flow and the gas flow is the potential for a hydraulic jump and the analogous potential for a shock wave (I think). --------- I think the 1/4 wavelength relationship for damped Rossby waves relative to topographic forcing might be specifically for the resonant wavelength; other wavelengths near the resonant wavelength might tend to be close to that phase relationship, etc... (?). -------- Nonlinearities II: breaking waves Waves of significant amplitude can become nonlinear and break. For example, for surface waves on water ... BACKGROUND there is a spectrum of gravity waves; waves with wavelengths much shorter than water depth are called deep water gravity waves; they are dispersive. They decay in strength over the depth of the water because the vertical accelerations of the up-down motion of any one layer partly cancel the effect of water level on pressure, thus reducing the forces that accelerate the water in deeper layers. Waves with wavelengths much longer than water depth are called shallow-water gravity waves. They are nondispersive - the phase speed c = w/k (thus w = c*k) is constant for all wavelengths, and thus the group velocity = del(w)/del(k) = c is the same as the phase speed (in the direction of phase propagation). They are approximately constant with strength through the depth of the water layer just because there is not enough depth for significant weakenning. Tsunamis travel through the open ocean as shallow water gravity waves because they have such long wavelengths. When any wave approaches shallower water, it slows down (for shallow water waves, c = sqrt(g*H); deep water waves are always slower because (I think) they don't 'feel' the full H). Thus waves refract toward shallower areas. The energy of the wave becomes concentrated into a smaller depth and also I think into a smaller wavelength (because the front of the wave reaches shallower water before the back and slows down first, I presume). Thus the amplitude increases.) (There is up-down motion of water as a wave passes (1/4 cycle out of phase with horizontal motion), as the surface moves up and down. This decreases with depth, because water at the bottom can only slide along it(except for the porosity and permeability of the bottom material, but let's set that aside for now) (although there is up-down motion associated with a bottom slope, but that's either in phase or 1/2 cycle out of phase with the horizontal motion). Thus the up down motion is associated with vertical stretching and shrinking, balanced and caused by convergence and divergence, which is forced by the spatially varying and cycling horizontal pressure gradient that is caused by gravity and the undulations of the water level at the surface (if the air had a larger density, the effect would be reduced; the pressure gradient per unit water level slope would be reduced and thus wave propagation would be slower, - the same as if the air were of insignificant density by comparison but if g were less - hence the use of a 'reduced gravity'. This is how internal gravity waves travel. The (internal) gravity waves in the atmosphere exist because of a more gradual stratification that can be quantified by N. Internal gravity waves can tilt, etc...)). Because of the back and forth motion, if the amplitude is large compared to wavelength, crests are sharpenned and troughs are broadened (I think). There's also something called Stokes' drift but I don't know much about it. Speed is faster in greater depth, but the crest of the wave has deeper water than the trough. This effect can be ignored for sufficiently small amplitudes (the anomaly water level is only used to compute the pressure gradient; otherwise only the basic state depth is used - an example of linearization). But for large amplitudes relative to water depth, the crest travels faster than the trough. Eventually the wave breaks. And there is surf. The atmosphere can also have surf (I've actually seen the term used in this context) from breaking Rossby waves. Unfortunately, no one has yet figured out how to hang 10 on a Rossby wave (a PhD thesis paper topic?). I have no idea what 'hang 10' actually means. (Eddies can be shed from meandering jets and currents - see Cushman Roisin p.250). -------------- Nonlinearities III: waves having waves, sharing energy Wave energy from one part of the spectrum can 'bleed' into other parts of the spectrum. There's something called nonlinear triad interaction/resonance??, but I don't know much about it (but it may have something to do with combinations of thee wave vectors which form triangles). This can happen with gravity waves and also I think Rossby waves. When a Rossby wave's amplitude is very large, conceivably the wave IPV gradient could become larger than the basic state IPV gradient. Because the wave IPV gradient reverses periodically across wavelengths, there is a potential for barotropic and/or baroclinic instability (see below), so maybe an intense wave can break down into smaller waves that way (?), but I'm not sure if this particular process is significant in the atmosphere. ---------------- Nonlinearities IV: geostrophic turbulence (see Cushman Roisin p.219-221, 257-261) Vortices of like sign tend to merge and under some conditions can survive for a long time (Great Red Spot - such a persisent state is prevented in the Earth's atmosphere and ocean by disruptive and dissipative forces of external origin relative to the fluid (Cushman-Roisin, p.261)). Vortices embedded in a basic state PV gradient that radiate Rossby-wave type disturbances tend to drift toward PV of the basic state of the same type (Cushman Roisin p.257-259 - PS on page 259, Cushman-Roisin mentions the "southeastward" drift of Hurricanes as being caused by this effect - I'm pretty sure that's a typo and it was supposed to be "southeastERLY" - as in "northwestward"). Perhaps this is also how and why vortices of like sign would tend to merge? Persistent strong nonlinear vortices may coexist with weak linear waves in between them (Cushman-Roisin, p.220-221). ___________________________ Barotropic and Baroclinic Instability: If there is a reversal of the IPV gradient somewhere, then there can be some combination of barotropic and/or baroclinic instability about the reversal. On one side of the reversal, the waves propagate one way, and the opposite on the other, if left to themselves; they thus propagate relative to each other. If there is a basic state wind shear, then the waves might be carried back - if one or both are propagating upstream, the wind shear might slow or reverse the propagation of one set relative to the other. The wind field from one wave can extend across the reversal; this wind field tends to produce a wave that is 1/4 wavelength out of phase, in the direction of free Rossby wave propagation on it's side of the reversal, from the waves on the other side. the wind field of the new wave will tend to produce a wave on the first side of the reversal that is out of phase in the same way, and that happens to be in phase with the first wave. Thus these two sets of waves can mutually amplify each other. If the two sets across the reveral are in phase or 1/2 wavelength out of phase, they do not amplify each other (or cause each other to decay) but they cause each other to propagate away from being 3/4 wavelength out of phase and toward being 1/4 wavelength out of phase. If the two sets are 3/4 wavelength out of phase, they cause the mutual decay of each other. So at anywhere except at 1/4 and 3/4 wavelength out of phase, but peaking at in phase and 1/2 wavelength out of phase, the waves act on each other's spaces to bring them away from mutual decay (strongest at 3/4 out of phase but occurs anywhere from 1/2 to 1 or 0 to -1/2, etc.) and toward a state of mutual amplification (strongest at 1/4 wavelength out of phase but occurs anywhere from 0 to 1/2 wavelength out of phase). The combination of different wave propagation directions and their speeds (which is largest for long wavelengths and large IPV gradients on either side with small S, etc., depending on whether it is more baroclinic or barotropic instability) and wind shear across the reversal, may continually tend to disrupt the phase alignment (unless they themselves cancel each other), but as the phase alignment shifts, the mechanism described above can change the phase propagation speeds - the amplification is not as rapid but the phase alignment can still persist - however, for some wavelengths and some conditions, it will be impossible; the necessary phase locking is prevented (See Bluestein p.207-211; Holton Ch.8; Cushman-Roisin p.250, Ch.16, Ch.7; Martin; Wallace and Hobbs, etc.). Of course, if the waves extend away from the reversal, the influence of the other waves is reduced and their own self-propagation dominates. This could result (before accounting for basic state wind shear) in a continuation of the tilt of the wave in the same sense as is found if connecting the phase lines (planes or surfaces in three dimensions) across the reversal. In that case, the group velocity is then directed away from the reversal in both directions, which makes sense since that's where the wave activity is being produced (but of course, carrying wave energy away will reduce the amplification at the source). Enhancing this tilting tendency is if the magnitude of the basic state IPV gradient continues to increase away from the reversal (where it was zero, unless there is some gradient along the reversal, but that gradient must be small in comparison to the perpendicular gradients on either side, I think). In order for the instability to occur, it must also be the case that somewhere (likely near or at the reversal), the air must be moving with the growing wave pattern and vice versa. Because the air (or water) in such a critical level is not moving through the wave pattern, the waves' wind fields continually deform material surfaces or lines in a one-way, non-cycling manner. In the case of a reversal in the horizontal (technically, horizontal along an isentropic surface for the IPV perspective), this requires an IPV maxima or minima at the reversal. This is barotropic instability. In this case, the deformation of IPV contours around the critical level is such that the band of maximum or minimum IPV (a sort of shear line), marking the reversal of the IPV gradient, breaks up into seperate vortices. I'm not sure but I think these might be called Kelvin's cat's eyes (??). Such a feature is also present in Kelvin-Helmholtz instability, which is a general phenomenon that can occur on various spatial scales with various orientations - vertical shear, which has to be sufficient to overcome vertical static stability where that occurs (sometimes absent in the boundary layer), has Kelvin Helmholtz instability that may be made apparent by billow(s?) clouds. In fact, this instability is at least partly responsible for the puffy texture of cumulus clouds (the air is rising (and may have different horizontal momentum) within the cloud, thus the edge of the cloud may have some maximum in wind shear). It is also interesting to note that, in so far as the basic state IPV gradient is from wind shear and not just variations in S and/or f, the tilting of a decaying barotropic Rossby wave (interpolating through the 3/4 out of phase relationship) is that which occurs when the basic state wind shear deforms a untilted barotropic Rossby wave - as I described somewhere above, the eddy momentum flux by waves tilted with the shear, the average across wavelengths of u'v', is such that the waves tend to concentrate momentum in the direction of the basic state shear - causing a jet to grow stronger. And the barotropic instability description suggests such a wave tilt may have some convergence of group velocity and thus wave energy toward the maximum in basic state shear, and the the waves decay. On the other hand, those waves which are tilted in the opposite direction will pull momentum out of a jet and across a basic state shear zone, tending to reduce the shear (averaged across wavelengths) - and these are the waves that could grow by barotropic instability if the propagation properties and conditions are right. (SEE also wave-mean interaction, coming up). This is actually analogous to the more familiar small scale turbulence in a shear zone - for example, the atmospheric boundary layer (the lowest part of the atmosphere, which exchanges momentum with the surface via friction and eddy fluxes). Shear causes eddies to grow; those eddies are tilted against the shear and on average transport momentum toward the surface. The basic state shear might concievable tend to tilt the eddies with the shear, so that they would decay and pull momentum back from nearer the surface (as if there were negative eddy-viscosity). However, on these scales, smaller eddies can produce yet smaller eddies, losing kinetic energy to eddy viscosity, and the smallest eddies easily lose kinetic energy to molecular visocity (Someone wrote a poem about this!). If the reversal occurs in the vertical, it is Baroclinic instability ...
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396. Patrick 027 at 14:51 PM on 11 January, 2009
     "(Eddies can be shed from meandering jets and currents - see Cushman Roisin p.250)." - and also, Bluestein p.211-213. "the eddy momentum flux " Another way (I think?) to visualize that is that when a wave is tilted, the wind coming around the bend has to leave the bend with some momentum that is different than it had before, and something has to provide the acceleration, and there must be some equal and opposite reaction somewhere (distributed, of course). Of course, the distinction between streamlines and trajectories alters this picture but I think in the case of the transport of u' by v' between shear of u in the y direction, it still works out... ------------- Barotropic waves: Wavelength dependence of instability - without counteracting wind shear, the longest wavelengths are the most likely to escape phase locking and not be amplified; For a given wind shear of the same sense as the IPV maximum or mimum, the shortest wavelengths will be the most easily pulled away from phase locking. However, the range of unstable wavelengths could be shifted a bit toward longer wavelengths because the wind fields of longer wavelengths penetrate farther across the reversal so that the waves on opposite sides can interact more strongly. Also, as wavelength increases, the phase speeds increase up to a point and then approach a maximum value due to the effect of divergence to maintain near geostrophy; so in some conditions, perhaps there is not a long-wave cutoff for instability. Of course, this is all altered by S, because AV = IPV/(S*g). Also, unless there is an actual gap with no IPV gradient in between two regions of constant IPV gradient, there is no obvious set distance to use between the two sets of waves, so it could be more complicated. Cushman-Roisin p.250 suggests critical wavelengths (I assume this means unstable or most unstable in this context) scale with the width of a jet. I'll have to go back to Ch. 7 to check out how that works. Bluestein p.211 suggests the midlatitude region in between the subtropical and polar-front jets (a relative minimum in westerly winds has anticyclonic RV on the poleward side and cyclonic RV on the equatorward side) could be a place where there is barotropic instability (taking into account beta (the gradient of planetary vorticity (f)), which is always increasing cyclonic vorticity poleward.). More wave-mean interaction: as the contours of IPV deform and the band of minimum or maximum IPV breaks up into vortices, there is a net (averaged across wavelengths) transport of IPV down-gradient - toward a minimum and away from a maximum in the basic state. ------------- Baroclinic Instability: -- (NOTE basic state is assumed to be near geostrophic balance or gradient wind balance - this implies that a vertical wind shear (typically, increasingly westerly (to the east) with height) is proportional to a horizontal temperature gradient (toward the equator for the case of the just-described westerly vertical shear). The temperature gradient, for other given conditions, is proportional to the slope of isentropic surfaces (in terms of isobaric coordinates in particular) multiplied by S (the static stability).) -- The IPV gradient being considered is in the 'horizontal' (actually along isentropic surfaces, which can and do slope where there is a horizontal temperature gradient). Thus there isn't generally an IPV maximum or minimum at a reversal of the IPV gradient over vertical distance. For a non-IPV non-isentropic (not in x,y,q coordinates but rather x,y,p) perspective, see (Quietman already saw this) comments 76 and 77 in particular at: http://www.skepticalscience.com/volcanoes-and-global-warming.htm (And also Holton Ch.8, and Martin, Bluestein, etc.) By some nice clear reasoning, Bluestein discusses how instability is related to wavelength and basic state characteristics on p.208-211 (p.210-211 in particular). Small S and large wavelength L increase the interaction of two sets of waves across a vertical distance (via the vertical penetration of an RV anomaly and hence the wind field from an IPV anomaly); S does this both by increasing the vertical spread of RV from it's maximum (at the level of it's source IPV anomaly)and the value of RV at all levels (because AV is proportional to IPV/S, etc...). However, these effects also increase the self-propagation speeds that would tend to pull the two wave sets out of any phase locking. Larger f (actually, larger basic state AV and f_loc) increases the interaction across a vertical distance. The interaction is stronger also if the vertical distance is smaller. Larger vertical wind shear (in the opposite sense of the vertical variation of phase propagation) can help overcome the effect of self-propagation. Bluestein concludes that there is a range of wavelengths that can amplify by baroclinic instability, and the range can shift toward longer wavelengths L if S is large and/or the basic state vertical wind shear is large (both effects make it harder for shorter wavelengths to be amplified). (Perhaps, since the effect of S on self-propagation has a proportionate effect on RV at all levels, the two effects nearly cancel (?) so that the effect on vertical spread of RV dominates (?)). However, this doesn't appear to take into account the relationship between the IPV gradients, S, and the vertical shear - although larger IPV gradients would enhance both self-propagation and wave growth, so perhaps the instability is not so sensitive to that. ---- ALTHOUGH, I wonder what happens if one IPV gradient is much weaker than the one across the reversal (for either baroclinic or barotropic instability) ---- **(PS note that propagation in the horizontal direction is actually, to be safe, along isentropic surfaces (for adiabatic motion), in the IPV perspective). But notice that in the above, the description seems to be of two single layers or surfaces with IPV gradients, rather than a horizontal IPV gradient that continuously changes over vertical distance. Indeed that is about the case. Although I'd guess analysis of the more widespread IPV gradient case would yield similar results. For the real troposphere, however, the case of two distinct surfaces with opposing IPV gradients actually works as an approximation. There are IPV gradients generally throughout the atmosphere, but the are generally weak in the bulk of the atmosphere; there is a relatively sharp increase in the IPV gradient along isentropes going across the tropopause and into the stratosphere; the gradient is generally increasingly cyclonic IPV toward the poles, which is dominant in the atmosphere as a whole. For reasons discussed in comment 313 above, the generally equatorward temperature gradient at the surface is such that the dynamics are as if there is an IPV gradient of increasingly cyclonic IPV toward the equator (this could be better understood with an approximation that isentropes' slopes decrease suddenly going toward the surface, and then continue equatorward just above the surface; the piling up of isentropes next the the surface results in a increasing S or increasingly thick layer of high S going toward the equator, and a sharp increase in IPV magnitude along isentropes upon entering the high S layer (that's the most important part of this, I think) - notice the similary of this situation to what happens crossing the tropopause along an isentrope (PS I think IPV surfaces can be used to define the tropopause at mid-to-high latitudes (while isentropic surfaces cross the tropopause), but at low latitudes, isentropic surfaces don't cross the tropopause so much while IPV surfaces do. Undulations of the tropopause are associated with IPV anomalies; a depression in the tropopause tends to be associated with a cyclonic IPV region). Thus, to a first approximation, the troposphere might be described in terms of generally opposite surface and tropopause IPV gradients - as if the three dimensional troposphere could be described by a two dimensional hologram! (But that is only an approximation). In fact, in the fully three dimensional description of baroclinic instability, describing what happens within the air, there is no IPV gradient necessary; baroclinic instability depends fundamentally on a temperature gradient and the coriolis effect (although beta and IPV gradients will modify behavior, they are not the fundamental reason for the existence of baroclinic instability). If there is no IPV gradient within the air, however, there will be an IPV gradient at the top and bottom boundaries because a temperature gradient implies sloping isentropes and these isentropes must hit the floor and cieling unless there is a layer of high static stability to collect them (which itself would tend to require an IPV gradient within the air). As I recall Holton's description (Ch. 8) suggests that a temperature gradient at the lower boundary (the surface) is necessary for baroclinic instability. But that seems a little odd - is there a way to have baroclinic instability away from the surface? (it can certainly be away from the top of the atmosphere). Well, it may not be likely because of the dominance of beta in IPV gradients within the air, and also that while there is westerly shear associated with the typical temperature gradients, it is generally westerly from the surface on up in the midlatitudes. ...
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397. Patrick 027 at 15:55 PM on 11 January, 2009
     ... In reality, the westerly jets that increase in strength toward the upper troposphere reach a maximum near the tropopause as a consequence of a general reversal of the temperature gradient going into the stratosphere (Although in winter, the temperature gradient is not reversed everywhere, and there is another westerly jet at higher levels at high latitudes (the polar night jet, I believe). PS the tropopause generally slopes downward toward the poles, with breaks at jets. (climatological seasonal zonal (over all longitudes) averages of the zonal wind shows only one westerly jet maximum in each hemisphere (outside the stratosphere) - but there can be two such jets at any one time and longitude: the polar-front jet is associated with strong thermal gradients generally at mid to high latitudes, while the subtropical jet is associated with thermal gradients over the subtropics in the upper troposphere that are not so strong at the surface (perhaps associated with converging meridional winds from the Ferrel and Hadley cells? - and also perhaps due to a reduced dry static stability (S) toward the tropics due to the effect of higher temperature on the moist adiabatic lapse rate). The subtropical jet's vertical wind shear can be stronger for it's associated thermal gradient because the coriolis effect (magnitude of f) is weaker at lower latitudes. Of course, at any one time, jets can merge and branch and break off into closed loops and have confluent entrance and difluent exit regions, etc.) Static stability, in winter in particular, increases within the lower troposphere from midlatitudes toward polar regions; geometrically this allows the equatorward temperature gradient to decrease with height. ... Imagine instead the case of an easterly jet at some level, with a westerly jet at some level above it, implying an equatorward thermal gradient in between. No such gradient (or a weaker or opposite gradient) below requires an increase in S toward the equator at the level of the easterly jet (although just having a much reduced gradient could be accomplished just by a reduction of S going downward). The easterly (= westward) jet has anticyclonic RV on the poleward side and cyclonic RV on the equatorward side. These combined may be able to overcome the beta effect, so that there could be an equatorward cyclonic IPV gradient at the level of the easterly jet. Then there could be baroclinic instability away from the surface. Actually, it need not be an easterly jet; it could just be a relative minimum (in the horizontal) in the westerlies. Moist processes: Cyclones generally have precipitation and latent heating, so they are not dry adiabatic; isentropes are not reliably material surfaces even if radiational heating and cooling can be set aside for short periods of time. What could be used instead where latent heating occurs are surfaces of equivalent potental temperature (pseudoisentropes?), and the Rossby waves within such regions would depend on arrangements of equivalent-IPV. Alternatively, one can account for the creating and destruction of IPV by diabatic processes (and viscous processes, too, as long as we're at it). IPV is generally created beneath a maximum in heating and destroyed above it (while diabatic heating will transport it upward across isentropes). (The relationship is not 100% precisely so because the same heat causes essentially the same change in Temperature T at all levels, but the change in q, while proportional to T, also depends on pressure p. An evenly vertically-distributed increase in T would cause greater increases in q at lower p, thus reducing S). Using the IPV perspective may help in trying to understand how climate change would alter storm track activity (again, see also comment 76 at http://www.skepticalscience.com/volcanoes-and-global-warming.htm ). __________________________
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398. Patrick 027 at 16:44 PM on 11 January, 2009
     Fronts - Fronts can be/are very important mesoscale aspect of baroclinic waves (in extratropical cyclones in particular) and jets. The most familiar fronts (seen on TV weather maps) are strongest at the surface, but some upper-level fronts occur, too; I think they are associated with jets; they can be extrusions of stratospheric air into the troposphere (frontal zones have higher S, although they may occur in larger environments of relatively smaller S). Fronts can develop from quasigeostrophic mechanisms, but there are faster mechanisms (with greater ageostrophic effects) that allow faster frontal development - there is a kind of instability in that a strong temperature gradient can act to intensify itself (but quasigeostrophic processes can and do produce the seeds of fronts). In the vertical, frontal surfaces (at least the lower level ones or the portions within the troposphere (?) slope over their colder sides going upward. Air trajectories don't cross frontal surfaces much (at least for lower level fronts (?)), so fronts generally move with the wind. ... I'm not going to spend more time on Fronts here. --------------- While a fixed forcing may produce some Rossby wave whose phase lines are stationary, the wave activity can spread with the group velocity. For a northward vorticity gradient, the group velocity, depending on wavelength and direction, can be to the east or west, but it can never keep up with the phase lines in their direction of propagation relative to the air as blows through them. Thus, wave activity spreads downstream along a westerly flow; westerlies continue to meander after passing just a single forcing (mountain range, etc.). In contrast, easterlies can (at least in an idealized setting) respond with just one displacement at the forcing and resume straight flow. **Extensive teleconnection patterns due to the forcing of Rossby waves by, for example, deep convection over a tropical SST anomaly (ENSO or what-have-you), require (as far as I know) that the winds in that region are westerly (to the east). ______________________ Wave Mean interaction: If the basic state has a constant IPV gradient and the Rossby waves are constant in amplitude along the gradient, then, the average IPV over an integer number of wavelengths is unchanged by the presence of the wave. If, however, this is not the case - for example, if the Rossby wave amplitude decreases or increases along the IPV gradient, then the average over wavelengths of the IPV can change. There will generally be a net shift of IPV down gradient, as if the Rossby waves have mixed the IPV (although without wave breaking or some other things, the mixing is reversable because there has been no mixing on the scale of the waves). In so far as this contributes to a net change in RV, this means the presence of the wave has caused an average increase in RV toward lower original IPV and the opposite in the opposite, which thus requires some change in the average wind in between. This is related to the EP flux of the waves. It makes sense that an average over wavelengths of vorticity flux (i.e. or e.g. - average of v'RV') must occur with convergence or divergence of the momentum flux (u'v'), because concentrating momentum in one place tends to produce a new RV' gradient across that place. vertical variation of thermal fluxes v'T'(vertically differential temperature advection) suggests a change in S. Considering averages over x and looking at the EP flux in the vertical plane y,z (taking the default of y being toward the north for convenience): The upward component of the EP flux is proportional to the average of v'T' - it is proportional to the eddy temperature flux (but also depends on N, H, f, and density - see Holton, p.323). Convergence of this component thus implies that stability S is increasing more to the south and/or decreasing more to the north, which suggests a decreasing northward IPV gradient. The northward component of the EP flux is proportional to the negative of the average of u'v' (and also density - Holton, p.323) - it is proportional to a northward eddy easterly (westward) momentum flux. Convergence of this component thus implies a decrease in the northward RV gradient, which would tend to contribute to a decrease in the northward IPV gradient. Of course, each variable is not seperately conserved - but in the process of geostrophic adjustment, the effect on IPV should be at least qualitatively the same, and Holton p.326-327 proves mathematically (not that I was able to follow every step, but at least it makes sense qualitatively) that a southward eddy flux of IPV (multiplied or divided by constants ** so as to be the quasigeostrophic potential vorticity, given in units of vorticity) is equal to the convergence of the EP flux, divided by density. Given the prior description of the effect of a maximum in Rossby wave amplitude, for a basic state northward IPV gradient, EP flux convergence should correspond to increasing wave amplitude, while divergence should correspond to decreasing amplitude. Which suggests that the wave energy and activity moves with the EP flux.
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399. Patrick 027 at 13:12 PM on 12 January, 2009
     EP flux clarification (Holton p.323): EP stands for Eliassen-Palm The y component: - (u'v' averaged over x) * density The z component (R and H in particular are not the same R and H I had referred to earlier.): (v'T' averaged over x) * density * f0 * R / (N^2 * H) I think f0 is a representative f over a band of latitudes; the variation in f is taken into account in another way using a quasigeostrophic approximation. In this context, H and R are NOT than the height scale of a particular phenomenon and the Rossby radius of deformation. H is the scale height of the atmosphere, which is the height over which density and pressure decrease by a factor of e (Holton, p.253 - in actuality these can be different and H can vary over height and space due to temperature variations, but the purpose here is to use "log-pressure coordinates", where vertical distance is "z*" - where z* = H * ln(p/ surface pressure) (ln = natural logarithm) which is an approximation of the actual geometric height z. R is just the ideal gas constant given in terms of mass rather than moles (which means it varies depending on composition). I had initially described the EP flux convergence as being in the Northern Hemisphere; the point here is that EP flux convergence should always imply a net flux of IPV down gradient (at least where RV variations are not significant relative to beta - I think). Note f and typical IPV values are negative in the Southern Hemisphere; in the Southern Hemisphere, cyclonic RV is negative... _____________________ Anyway EP flux divergence acts to increase the zonal average zonal (westerly) wind speed, while EP flux convergence slows it down or makes it more easterly, which makes sense given the change in IPV distribution it is associated with.
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400. Wondering Aloud at 13:37 PM on 12 January, 2009
     I just came here because there were so many posts but Phillipe I could not disagree more with you on your definition of what a scientist is. A scientist is someone who uses scientific method. Compared to this one requirement, advanced degrees and "expert" recognition are just trivia. Also how often a paper is cited is not an indication of accuracy though it may be an indication of popularity. At best it may tell you who had an idea first. If you want an ugly experience spend a few days tracing references in papers and find out how many are fake, misquoted, unrelated or describe other "experiments" so badly done they were meaningless. See how long it takes you to find a circular citation. (A cites B who sites C who was citing A in the first place) In some fields 50% bs is fairly common, I've seen PhD disertations from well known "experts" that when you eliminate the fake references and the unrelated ones didn't have anything left.
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