P. SIMLATIOS OF ITERAL WAVES APPROACHIG A CRITICAL LEVEL Brian P. Casaday and J. C. Vanderhoff Brigha Young niversity, Provo, tah. BACKGROD Internal gravity waves exist abundantly in our world in stably-stratified fluids, such as the ocean and atosphere. A stably-stratified fluid is one where the density of the fluid continuously decreases with increasing elevation. When a finite aount of fluid fro a stratified environent is reoved fro its equilibriu density location by oving it upward or downward, buoyancy forces will copel the fluid to return to its original location. If the fluid is forced to oscillate at any frequency below the buoyancy frequency, it ay oscillate in the horizontal direction in addition to the vertical direction. These ultidiensional oscillations evolve into internal gravity waves, which will propagate throughout the fluid lie regular surface water waves, but different in that they ay propagate in three directions, and are not restricted to the horizontal plane of a liquid - gas interface. The wave phase and group speeds propagate orthogonally. pollutants, organiss, and heat in the ocean and atosphere, and transport oentu. Figures and illustrate how gravity waves ay becoe visible in our atosphere. As the fluid oscillates, water condenses and fors visible clouds in the crests of waves. Figure Internal gravity waves propagate in a stratefied fluid, soeties leading to visible wave clouds. www.weathervortex.co These waves are created through disturbances or perturbations in any stratified fluid, which occur regularly through natural forces, such as air or water currents oving over topography, interactions of fluid flows, or the breadown of large waves into saller scale waves or turbulence. Internal waves range in size, with wavelengths easured in scales of tens or hundreds of iloeters, and as such carry significant inetic energy. These waves significantly affect the dynaic flows of the ocean and atosphere. They contribute to the ixing of Corresponding author address: Brian P. Casaday, Brigha Young niversity, Departent of Mechanical Engineering, Provo, T, 8464 8-4-3 briancasaday@gail.co J. C. Vanderhoff, Brigha Young niversity, Departent of Mechanical Engineering, Provo, T, 8464 jvanderhoff@byu.edu Figure Gravity wave clouds fored by winds over Asterda Island. visibleearth.nasa.gov
The scales of practically applicable flows in the atosphere range in size fro iloeters to illieters or saller. This creates a series of probles regarding observational data collection as well as resolution of nuerical odels. Due to coputational cost, nuerical odels are often siplified by adopting linearized equations or twodiensional siulation. These siplifications result in liitations, in that with the onset of turbulence, linear odels brea down and twodiensional siulations lose validity. However, siplified odels are usually preferred when nuerous flows are analyzed. appo () explained that linear siulations were generally adequate and often preferred for any internal wave siulations in the iddle and upper atospheres. Of particular interest are interactions of internal waves with other geophysical flows, because of their regular occurrence in the environent. Winters and D Asaro (989) nuerically studied the interaction between sall scale internal waves and a strong steady shear, where the bacground wind continuously increases with altitude. The result is a critical level, where the phase speed of the wave atches the ean velocity of the bacground flow, and the relative frequency of the wave approaches zero. Critical levels are coon in natural environents, such as where a wind blows above a region of topography where internal waves are generated. This point represents an iaginary boundary where instability is liely to occur, and critical level theory predicts that internal waves are incapable of propagating past this point. The bacground shear, or the rate of horizontal velocity change with elevation, causes the direction of the wave propagation to becoe ore horizontal as it approaches the critical level. Winters and D Asaro showed that if the aplitude of the internal waves were sufficiently sall, energy fro the waves approaching a critical level was lost to the bacground flow, accelerating the flow near that region. If the aplitude of the waves were larger, the waves would steepen and could potentially overturn and brea, as discussed in later wors of Winters and D Asaro (994). As these waves would steepen and overturn, analogous to water waves approaching a shore, energy was lost to turbulence and viscous dissipation. Perturbations fro the overturning waves resulted in the foration of new, saller internal waves that propagated past the critical level. Soe waves were reflected fro the critical level and propagated away in the direction fro whence they appeared. Thorpe (98) researched this interaction experientally and found that wave behavior was surprisingly accurate for his nuerical odel, for which later researchers copared and validated their results. Linear odels are advantageous for critical level siulations, due to significantly increased coputational speed. Winters and D Asaro showed that two-diensional siulations are adequate prior to wave breaing and turbulence. This research is helpful in understanding large scale energy budgets and nowing where the energy is distributed or lost. Steady critical levels have been extensively studied, yet less research has been done with tie dependent critical levels, which are ore prevalent in the environent.. METHODS A linear ray-tracing progra was used to siulate the interaction of internal waves approaching a critical level, as well as waves approaching a tie dependent critical level and a large-scale inertial wave. The ray tracing progra was written in MATLAB and used an R-K 4 ethod for solving differential equations. The progra uses linear WKB theory to calculate the dynaic properties of the interacting waves, and used the following assuptions: The interaction of the sall wave does not affect or change the bacground winds. Buoyancy frequency is constant and average fluid density is treated as constant. Fluid viscosity is neglected. The WKB approxiation is valid prior to wave breaing or inflection. Many inputs are taen into the ray-tracing progra, and for the atospheric siulations discussed in this paper, any of the values were ept constant such as the following variables. These were the sae variables used by other authors. (Sartelet 3) (Vand 8) Coriolis frequency at id-latitude: f =.t - Buoyancy frequency: =. t - Density of air at about 5-3 altitude: ρ =.34 g/ 3 Density gradient approxiated fro RLMSISE Standard Atosphere Model
dρ =.386ρ dz For two-diensional waves, the wavenubers are defined as: = π λ V = π λ. Mid-frequency Approxiation Assuing >> ω >> f, which is nown as the id-frequency approxiation, the WKB equations siplify to: f ω = ± ω ± H / ( f ) ( f ) / ( ) 3 / c dz ω = = = gz dt ± c gz For a wave passing through a steady bacground flow with a differentiable flow field and constant,, and f, and initial wavenuber, each value of horizontal velocity throughout the field produces a distinct wavenuber. A critical layer is an area where the phase speed of the sall wave atches the ean bacground velocity, and the relative frequency goes to zero, or f if the fluid is rotating, such as the earth. This is defined atheatically as: ω = f = / Siplifying the above equation using the idfrequency approxiation results in: Fro these siplified equations, we can deterine the approxiate value of at any point within the bacground flow given the horizontal velocity, and it is shown that for a given velocity : CL c When =, then =, so c = : = ( ) This siple odel is valid for values where the buoyancy frequency is uch larger than the Coriolis frequency (which is generally true for any stable region of the Earth s atosphere) and when the value of is greater than. The following figures show the error associated with the idfrequency approxiation. Error %.% 5.%.% 5.% -.% -5.% -.% Siplification Error f =., =..% 4 6 8-5.% / Figure 3 Error associated with the id-frequency approxiation. Typical errors in the id-frequency range are less than 5%. ω cgz 3
The ray tracing progra does not use the idfrequency approxiation, but rather uses the full WKB approxiation. However, the id-frequency approxiation is useful in understanding the dynaics of the wave paraeters ore siply. The ray tracing progra easily calculates the wave action density, proportional to the aplitude A, and the wave steepness, using the following WKB equations: A = f ( f ) ( ) f = 3 ( f ) / ( ) z ς z = A = ωρ / 3/ z The local Richardson nuber is calculated to deterine instability due to shear forces, this is defined as: = d dz And the wave induced shear: ' u z 3. SIMLATIOS 3. Mean Wind A = ωρ / f ω / As waves approach a ean wind, ray theory suggests that waves will not be able to pass through the wind if a critical level is present. At the point where waves pass the critical level, ray theory predicts an infinite action density as well as zero vertical group velocity. For this reason, the waves will not actually reach the critical level, but rather approach it asyptotically, with its action density and wave steepness approaching infinity. If a critical level is not present, the waves will pass through the ean wind. The action density of the waves and wave steepness will increase as the waves approach the axiu velocity of the ean wind, or when the wave is passing through a positive velocity gradient. The wave steepness and action density decrease as the wave passes through the negative velocity gradient, or shear. If the conditions on either side of the wind layer are identical, then the conditions of the sall wave, including wave nubers, action density, steepness, and frequency, will all return to their original values. The wave steepness and action density reach a axiu as the waves pass through the axiu velocity. There is a solid relationship between the wave steepness, action density, and the ratio of the axiu wind velocity to critical layer velocity. The equations are shown below, the first confired by the idfrequency approxiation and then all confired with the ray tracing progra. Wave action density: A A Wave Steepness: ς z ς z ( ) ( ) ( ) 3 / These relationships are valid for different values of the wave nuber, axiu wind velocities, and wave nuber ratios /, provided that the ratio is larger than., with larger ratios fitting the curves ore tightly. Figures 4 and 5 show two siulations of sall waves approaching a ean wind, the first with no critical level present and the second with a critical level present. The wave that doesn t approach a critical level never becoes unstable, but the wave that approaches a critical level becoes unstable near the critical level. CL 4
z(t) Ray paths with Critical Level regions: = -4 =.683 - Initial Inertial-Wave Profile, (/sec) λ I = 5 5 (t=) V(t=) Critical Level Velocity z () z () -5-5 3 4 5 6 Wave Steepness =. / nstable >..4.6.8 /s Gradient Richardson uber 3 Bacground Short Wave nstable < /4 - - 3 4 5 6-3 4 5 6 Figure 4 Wave approaches a ean wind with no critical level. The wave passes through the ean wind, and the steepness of the wave increases during the interaction, but returns to its original conditions as the wave exits the wind. pper left: wave rays plotted in elevation versus tie. pper right: Initial bacground velocity profile. Lower left: Relative and actual wave steepness of first ray versus tie. Lower right: Gradient Richardson nuber versus tie for bacground with (and without) induced shear of sall wave. The tie is nondiensionalized against an inertial wave period, defined as T i = π/f z(t) Ray paths with Critical Level regions: = -6 =.683 - Initial Inertial-Wave Profile, (/sec) λ I = 5 5 (t=) V(t=) Critical Level Velocity z () z () -5-5 3 4 5 6 Wave Steepness =. / nstable >..4.6.8 /s Gradient Richardson uber 3 Bacground Short Wave nstable < /4 - - 3 4 5 6-3 4 5 6 Figure 5 An internal wave approaches a critical level and the vertical group velocity decreases asyptotically. The steepness increases and the wave becoes unstable. 5
3. Moving Mean Wind An understanding of the ean wind scenario provides the nowledge necessary to understand a wind envelope oving in the vertical direction, siilar to the phase speed of an inertial wave. This provides a step to understanding the interaction between a sall-scale wave propagating through an inertial wave. In this scenario a sall scale internal wave approaches a Gaussian envelope of wind, just lie in the previous scenario. However, this envelope of wind, although it has no vertical coponent to its wind velocity, oves as an envelope toward the approaching wave at a constant speed. In this way, the wave not only approaches the critical level, but the critical level then oves through it. Figure 6 shows the siulation of a wave passing through a oving ean wind. Fro this and other siulations it is concluded that once the pacet of wind has oved through the wave, the wave paraeters return to their original values before they had approached the pacet. This scenario, although not necessarily prevalent in the real world, leads to a better understanding of the interaction with inertial waves, where ultiple pacets of wind will pass through an internal wave. The variable paraeters in this scenario are,,, and M, referring to the speed of the wind pacet. We use the ter M because in the inertial wave interactions, the wave nuber M and the Coriolis frequency deterine the speed at which the wind pacets, or wave phases, will pass through the wave. In the previous scenario we saw that was dependent on the local bacground velocity, and the sae is true in this scenario, only that a new variable M is added to the equation. To find the changing wavenuber : = z(t) Ray paths with Critical Level regions: = -5 =.683 - Initial Inertial-Wave Profile, (/sec) λ I = 5 5 z () z () -5-5 (t=) V(t=) Critical Level Velocity 3 4 5 6 Wave Steepness =..5.5 /s Gradient Richardson uber 3 / nstable > Bacground Short Wave nstable < /4 - - 3 4 5 6-3 4 5 6 Figure 6 Wave approaches a oving ean wind. Although a critical level is present, the wave passes through the critical level region and returns to its original paraeters. The wave behaves as if it isn t even approaching a critical level. 6
± For a downward oving bacground wind, we can siply add the speed of the bacground wind envelope c to the vertical velocity of the approaching wave. Also, the bacground velocity can be nondiensionalized against the critical layer velocity, where ( f ) ( ) / = = ω / c ± Cobining the two allows us to find the change in wavenuber with changing velocity = c c c c sing the condition that = at =, we find that: Cobine to yield: C c C = c c ( ) c The vertical velocity that coes fro the downward propagating bacground wind can be related to the downward propagating phase speed of the inertial wave, which is f/m, where M is the vertical wavenuber of the inertial wave. Cobining these with the equation above we get: CL CL f M f M ( ) ( ) Fro the above equation, as becoes large relative to its initial value, it increases linearly with bacground velocity. As the wave passes through the pacet of wind (or the wind passes through the wave) continually increases until it passes the pea wind velocity. After which it decreases in the negative bacground shear and returns to its original value. In the previous scenario, where the bacground did not change with tie, the wave action density, directly related to the wave aplitude, would increase and decrease with an increasing or decreasing wavenuber. However, with oving critical levels, the wave action density reaches a axiu value near a velocity at about twice the critical layer velocity. This is shown in figures 8a-8d on the following page. All action density figures appear siilar, though not necessarily with the sae length scales. As different siulations were run with a different inertial wave nuber M, the action density versus the velocity increased linearly. Siilarly, as the wavenuber of the short wave is increased, the change in wave action density is linearly decreased. This fully agrees with the previous scenario of the waves approaching a ean wind. In those cases, the speed of the envelope of wind is zero, which corresponds to an infinite value of M. In these cases, the wave does not pass through the wind when the bacground velocity is 7
greater than the critical layer velocity, and the action density approaches infinity at the critical layer. In addition to the wave action density, it is iportant to now the wave steepness, which will cause the internal wave to theoretically overturn when the steepness is equal to unity. (Vand 8) Steepness plots are shown in figures 7a-7b. Many tests were run with oving envelopes of wind, and qualities were found. As entioned before, the action density increases with larger values of M, as well as saller values of. This is true when the bacground velocity is significantly larger than the critical layer velocity. For velocities uch lower than the critical layer velocities, the action density diinishes. Theoretically, the action density and steepness fro this oving wind should be a good estiate of the action density and steepness fro an interaction with an inertial wave. Figure 7 The wave steepness depends on the wave action density and the frequency of the wave. The steepness is therefore ore dependent on the wavenuber than the speed of the oving wind pacet. Figure 8 The wave action density increases based on the relative bacground velocity and speed of the oving wind pacet. During an interaction, the wave action density doesn't increase uch with winds greater than two ties the critical level velocity. 8
3.3 Reverse Mean Wind When a sall aplitude wave interacts with a large inertial wave, not only will it encounter oving critical levels fro envelopes of positive horizontal wind, but also envelopes of negative horizontal wind. If the negative wind is sufficiently strong, it ay cause the wave to change its vertical group speed direction. If the wave is able to propagate past the region of inflection, the vertical wavenuber becoes iaginary and the disturbance decays exponentially. (PedlosyThe inflection occurs when the frequency of the sall wave atches the local value of. By the dispersion relation: f ω = / It can easily be shown that when =, then ω =, and as approaches, then ω approaches f. ω = c gz The point at which the frequency coes closest to the buoyancy frequency is the point at which changes signs. The chance of inflection increases with increased axiu negative bacground velocity, larger, saller, and saller wavenuber M of the inertial wave. 3.4 Inertial Gravity Waves Inertial gravity waves are internal waves which exist in any rotating fluid and have a frequency close to the Coriolis frequency. At to 5 above sea level. R. O. R. Y. Thopson (978) showed that inertial waves are not present in the troposphere, but are coon in the stratosphere though their source often coes fro below. Sato, O Sullivan, and Dunerton (997) found inertial gravity waves over Japan with a period of about hr, and are doinant at an altitude of about, where bacground winds are sall. Guest, Reeder, Mars, and Karoly (999) deterined the properties of internal waves in the stratosphere south of Australia. Sato, O Sullivan, and Dunerton detected inertial gravity waves 4 over Shigarai, Japan which had wind oscillations of.5 to 3 /s. All sources found siilar properties of inertial waves, with a vertical wavelength between and 7, and horizontal wavelengths around. The frequency of the waves are near the Coriolis frequency, and the phases of the waves propagated downward. 3.5 Inertial Wave interaction Many experients were run where sall internal waves propagated through an inertial wave, with the inertial phases propagating downward. The inertial waves were given a vertical wavelength fro.5 to 7, with an infinite horizontal wavelength. As the wave propagated through phases where the bacground velocity is positive in the horizontal direction, the wave steepness and the wave action density increase, while the vertical group velocity decreases. As the sall waves propagate through phases of negative velocity, the steepness and action density decrease, while the vertical group velocity increases. Excepting cases of inflection or overturning, the steepness, action density, and group velocities return to their original values once the sall waves have propagated through the inertial wave. This assues that the bacground wind velocities are identical on either side of the inertial wave pacet. With soe sall variation, the wave action density and steepness are siilar to the scenarios of the oving wind envelopes. The variation arises fro the non unifor phases within the inertial wave envelope. Wave instability arises when the steepness of the wave exceeds unity, or when the bacground shear causes the local Richardson nuber to decrease below a value of.5. (Winters, D Asaro 994) Figures 9- illustrate two unstable interactions resulting in instability principally fro shear and steepness, respectively. 9
Figure 9 An internal wave interacts with an inertial wave. The lower plot shows an illustrated instability approxiation using the wave steepness and local Richardson nuber. Orange and red regions represent areas of instability.high bacground shears cause the sall scale wave to becoe unstable, even though the wave steepness reains low. z(t) Ray paths with Critical Level regions: = -4 =.683 - Initial Inertial-Wave Profile, (/sec) λ I = 5 5 (t=) V(t=) Critical Level Velocity z () z () -5-5 3 4 5 6-3 - - 3 /s Wave Steepness =. / 3 Gradient Richardson uber Bacground nstable > Short Wave nstable < /4 - - 3 4 5 6-3 4 5 6 Instability = /4, ζ =. 8.5 ray 6 4 3 4 5 6.5 Figure 7 An internal wave interacts with an inertial wave. The wave becoes unstable due to excessive wave steepness.
4. COCLSIOS Tie-dependent critical levels affect sall scale waves differently than stationary critical levels. Although critical level regions ay be regions of instability, they do not peranently affect internal waves as uch as stationary critical levels. Specifically, tie-dependent critical levels decrease chances of wave instability during the interaction copared to stationary critical levels. Wave instability, which ay lead to breaing, arises fro bacground shear and wave steepness, although one ay doinate the approach to overturning. Detected inertial waves in the atosphere typically do not produce low Richardson nubers (high shears) capable of instability. If wave instability arises during an inertial wave interaction, it is typically due to an excessive wave steepness of the approaching wave. This is dependent on the initial wave steepness before the interaction. Waves propagating through larger inertial waves are less liely to becoe unstable and brea copared to waves approaching stationary critical levels. 5. FTRE WORK These scenarios will be validated against nonlinear siulations or observational data. Other paraeters ay be necessary to aend the raytracing progra to increase its accuracy. Wave energy paraeters will be incorporated into the siulations. It is of particular interest to deterine how uch energy is transported or lost during an interaction. The ray-tracing progra will be enhanced to calculate the energy loss due to wave breaing and energy loss due to transfer to the ean flow. 6. ATICIPATED COTRIBTIOS This research on internal waves has any large scale global applications, and an understanding is necessary in creating global eteorological odels. As internal waves brea, they becoe a significant source of ixing within the iddle and upper atosphere, influencing the aount of pollutants or organiss in any given area. Internal waves aintain environental energy budgets by transporting ass, oentu, and heat throughout the global spectra. Because these waves are found in abundance over vast volues of space, uch energy is constantly being deposited or accuulated due to their propagation and breaing. An understanding of these processes is necessary to understand internal waves effect on global processes. 5. REFERECES Guest, Reeder, Mars, and Karoly, () Inertia-Gravity Waves Observed in the Lower Stratosphere over MacQuerie Island, J. Atos. Sci., 57, 737-75. appo CJ () An Introduction to Atospheric Gravity Waves, Vol.85 of International Geophysics Series. Acadeic Press, San Diego, 76 pp Pedlosy, Joseph (3) Waves in the Atosphere and Ocean, Springer-Verlag Berlin Heidelberg ew Yor Sartelet, K.., (3) Wave Propagation inside an Inertia Wave. Part I: Role of Tie Dependence and Scale Separation. J. Atospheric Sciences, 6, 448-455. Sartelet, K.., (3) Wave Propagation inside an Inertia Wave. Part II: Wave Breaing. J. Atospheric Sciences, 6, 448-455. Sato, O Sullivan, and Dunerton, (997) Low-fequency inertiagravity waves in the stratosphere revealed by three-wee continuous observation with the M radar. Geophysical Research Letters, 4, 739-74 Thopson, R. O. R. Y., (978) Observation of inertial waves in the stratosphere. Quart. J. R. Met. Soc., 4, 69-698 Thorpe, S. A. (98) An Experiental Study of Critical Layers. Journal of Fluid Mechanics, 95, 3-344. Vanderhoff, J. C., K. K. oura, J. W. Rottan, and C. Macasill (8), Doppler Spreading of internal gravity waves by an inertia-wave pacet, J. Geophys. Res. 3, C58, doi:.9/7jc439 Winters, Kraig B. & D Asaro, Eric A (989) Two-Diensional Instability of Finite Aplitude Internal Gravity Wave Pacets ear a Critical Level. Journal of Geophysical Research, 94, 79-,79. Winters, Kraig B. & D Asaro, Eric A (994) Three-Diensional Instability Wave Instability ear a Critical Level. Journal of Fluid Mechanics, 7, 55-84.