The significance of the nontraditional Coriolis terms - Hing Ongs PhD Dissertation Defense

So welcome everyone to Hing Ong’s dissertation defense Hing came to us with a Master of Science degree from Taiwan And he didn’t start out initially in my group I knew of him only from his original application to the department And just before he started in my in my group I woke up around 2:00 a.m. with my phone buzzing me for incoming emails And I don’t always look at these But I popped up and turned it on curiously I found him asking me to advise him on his PhD in tropical large- scale dynamics At the time, as I said, I didn’t know him very well But I’d seen his application And I knew he was a fantastic student with a great record In the months following Hing was, I think, the fastest student I’ve ever had in terms of turning out papers from the start of joining the research group He, as I said, he did have a master’s degree to start with But he’s always been quick studying And works really hard oftentimes in the middle of the night to get things done So it’s the first time obviously we’ve done dissertation defenses in this way The first part is as it’s always been the public portion Hing will provide his presentation for us And I ask that you please hold your questions until the end If you have questions that are brief clarifications then you address them to me in the chat And so I’ll be able to take a look at them and then assess whether either I can respond myself with a quick clarification or suggest that you wait to the end or I will step in for a moment and ask Hing to address the question depending on the nature of it Please if you can wait till the very end with those So we plan on the presentation taking roughly 45 minutes We’ll have about 15 minutes of questions from the public after which we’ll dismiss everyone but the committee The committee consists of myself, Brian Rose, Rob Fovell, and our visitor from NCAR Bill Skamarock And we specially appreciate his participation here And with that, we’ll let you take off Thank you, Paul It’s my pleasure to present my dissertation on The significance of the nontraditional Coriolis terms in tropical large-scale dynamics For those calling in while viewing my offline slides I will say the present slide number out loud Slide number 2 Like many other atmospheric scientists I used to believe that in our weather and climate models large-scale dynamics are well represented because we build our models upon equations known to govern the motion of large-scale flow And what needs improvement is to represent how the statistics of physical processes respond and feedback to large-scale flow For example, we need to improve convective parameterization which represent the statistics of convective processes So I used to study convective parameterization And I had a paper published about it three years ago That was right before I came to SUNY Albany However, two years ago when I was looking for a new research topic I reconsidered this question Are large-scale atmospheric dynamics well represented in our models? The result of this dissertation suggests slide number 3, no because the nontraditional Coriolis terms are omitted in most of the current global atmospheric models The nontraditional Coriolis terms are terms in the momentum equation that turn eastward motion upward and upward motion westward And vice versa They are usually omitted However this dissertation and a few previous papers suggest that the nontraditional Coriolis terms are considerable in tropical large-scale dynamics in terms of these three aspects (First,) flow response to heat sources Second, hypsometric relation or pressure-height relation Third, equatorial wave propagation mechanism The results encourage restoring the nontraditional Coriolis terms into the models for more-accurate simulations of MJO, the Madden-Julian oscillation CCEWs, the convectively coupled equatorial waves and flow response to the ITCZ flow forced by the ITCZ, the intertropical convergence zone You may wonder what made me consider such a question Slide number 4 I started from examining the equations what’s shown here are the fully nonhydrostatic momentum equations used by only few models including UK Met Office’s Unified Model, Japanese NICAM, and US Navy’s NUMA The yellow terms here Yellow terms are the nontraditional Coriolis terms They have a factor of 2 Omega cosine latitude which is associated with the meridional projection of planetary vorticity And they are maximized at the equator Then, slide number 5 These are the quasi-hydrostatic momentum equations used by earlier generations of UK Met Office’s Unified Model Compare slide number 4 and 5 The quasi-hydrostatic equations omit the vertical acceleration terms which makes sense because in 1995 White and Bromley’s scale analysis suggests that the vertical acceleration term is the smallest term in the vertical momentum equation for large-scale flow Then, slide number 6 These are the hydrostatic primitive equations used by most of the current global atmospheric models Compare slide number 5 and 6 Notice first here and here. Note that the vertical momentum equation reduces to the hydrostatic equation while one of the nontraditional Coriolis terms is omitted Then the other nontraditional Coriolis term in the zonal momentum equation must be omitted for energy conservation So to use the hydrostatic approximation is one of the reasons that people omit the nontraditional Coriolis term which makes sense because the balance between the perturbation pressure gradient and the buoyancy is the leading order balance in the vertical So far everything makes sense Slide number 7 These are the nonhydrostatic primitive equations used by many models on the pathway toward relaxing the hydrostatic approximation Compare slide number 6 and 7 Vertical acceleration term is restored But the nontraditional Coriolis terms are not So slide number 8 According to a scale analysis Vertical acceleration term is far smaller than the vertical nontraditional Coriolis term for a horizontal length scale of 100 km or larger However, in many models on the pathway toward relaxing the hydrostatic approximation the vertical acceleration term is included while the nontraditional Coriolis terms are omitted This inconsistency motivates me to ask this question Is there insufficiency in omitting the nontraditional Coriolis term? Before I move on, I reviewed some papers slide number 9 The general idea of these papers is to compare the nontraditional Coriolis terms to the leading order terms for scale analyses and to compare biases due to omitting the nontraditional Coriolis terms to results with or without the nontraditional Coriolis term for model simulations For tropical diabatic-forced large-scales flow White and Bromley’s scale analyses suggest that the nontraditional Coriolis terms are 10% of the leading order terms which are the traditional Coriolis in the zonal and the buoyancy in the vertical Hayashi and Itoh’s linear forced-dissipative model simulations suggest that the nontraditional Coriolis terms contribute 10% of the resulting wind and pressure Hayashi and Itoh’s forced So in their paper They force the model by a prescribed heat source that propagates eastward along the equator in an intraseasonal timescale Accordingly, slide number 10 In this dissertation, I further addressed these three questions In Chapter 2, how do the nontraditional Coriolis terms affect flow response to heat sources? In Chapter 3, how do the nontraditional Coriolis terms affect hypsometric relation or the pressure-height relation? In Chapter 4, how do the nontraditional Coriolis terms affect equatorial wave propagation mechanism? I’ll first show the flow response Slide number 11 I formulated a linear forced-dissipative model that can switch the nontraditional Coriolis terms on and off which is similar to Hayashi and Itoh’s model I forced the model with an ITCZ-like heat source which is steady and zonally symmetric And because it is steady and zonally symmetric I am allowed to make my model simpler than Hayashi and Itoh’s model And here shows the meridional vertical distribution of meridional velocity contoured and vertical velocity shaded As a sanity check the vertical velocity is nearly proportional to the prescribed heating rate So I put my heating rate here prescribed And there is an in-up-out Hadley-like circulation Then, slide number 12 Here shows the meridional vertical distribution of zonal velocity with the nontraditional Coriolis terms contoured So there are subtropical jet maxima here and here associated with the Hadley-like circulation the shading denotes the zonal velocity difference without the nontraditional Coriolis terms minus with So the shading can be interpreted as a bias due to omitting the nontraditional Coriolis terms Compare slide number 11 to 12 We see that a westerly wind bias is located in the heating region while the subtropical jet is not affected This westerly wind bias here is due to lack of westward nontraditional Coriolis term associated with heating-induced upward motion I’d like to reemphasize that the heat source is prescribed In this case, the ITCZ width is 1,000 km and the ITCZ location is 600 km from the equator If we divide the maximum westerly bias by the maximum westerly wind in the subtropical jet as a normalization we get 12% of bias Furthermore, slide number 13 How does the normalized wind bias change with the prescribed ITCZ the configuration? Here the normalized wind bias is plotted on the parameter space of ITCZ width and ITCZ location from the equator The plus sign here The plus sign here denotes what I just showed 1,000 km wide and 600 km from the equator And the normalized wind bias increases when the ITCZ is narrower or closer to the equator Inspired by these results Slide number 14 how hydrostatic is ITCZ-force flow? the traditional measure here to validate the hydrostatic approximation is to compare the vertical acceleration term to the buoyancy term The scale analysis suggests that the ratio is D-squared / L-squared where D denotes characteristic depth and L denotes characteristic width This ratio is very small for typical ITCZ which suggests that ITCZ-forced flow is in hydrostatic balance However, the hydrostatic approximation not only omits the vertical acceleration term but also omits the nontraditional Coriolis terms So an alternative measure for the tropical atmosphere is to compare the nontraditional Coriolis terms to the traditional Coriolis in the zonal momentum equation The scale analysis suggests that the ratio is aD / YL where “a” denotes planetary radius and Y denotes distance from the equator to the subtropical jet For a nonzero Y, we don’t consider any jet at the equator This ratio can be used to explain why the normalized wind bias increases when the ITCZ is narrower or closer to the equator And it is on the order of 10% for typical ITCZ which suggests that the hydrostatic approximation is associated with 10% bias in ITCZ-forced flow OK. Slide number 15 We’ve talked about how the nontraditional Coriolis terms affect how flow responds to heat sources Note that I prescribed the heat source as if I knew the distribution of the heating rate In reality, the heating distribution is difficult to simulate However, even if we had a perfect representation of the statistics of the physical processes which simulates the evolution of the heat source perfectly the forced flow would still be substantially biased if the nontraditional Coriolis terms are omitted With that in mind Let’s move on to the hypsometric relation or the pressure-height relation Slide number 16 I’ll start with a schematic Consider a zonal vertical space Given the pressure at the sea surface we can estimate the height of a pressure level aloft As a simple demonstration If the density is horizontally homogeneous the hydrostatic balance would suggest a flat isobaric surface aloft Then, the quasi-hydrostatic balance further introduces the nontraditional Coriolis term as a correction to gravity Westerly winds are associated with upward nontraditional Coriolis term which effectively makes the air lighter So it needs a greater thickness between pressure surfaces to balance The opposite is true for easterly winds Then, slide number 17 Here is the corrected hypsometric equation Most of it is the same as the traditional hypsometric equation It means that the thickness between two pressure levels, p2 and p1, equals to the integral of gas constant R times virtual temperature Tv divided by gravity But the gravity has a correction factor 1 + A where the correction A includes the nontraditional Coriolis term And it’s so-called the Eötvös effect The two nontraditional metric terms here are included for completeness But they don’t make a noticeable difference I calculated the height of pressure levels using this equation with and without the correction A using more than 300,000 selected tropical rawinsonde profiles And here, slide number 18, shows some of the results The figure shows the probability density of all available data points The vertical axis denotes the bias of height due to omitting the nontraditional correction at 500 hPa In terms of one standard deviation the range of height bias is roughly between 0 and half a meter And the height bias at 500 hPa is negatively correlated with the zonal wind at 700 hPa which is plotted on the horizontal axis With the linear regression An easterly wind of 8 m/s is associated with a positive height bias of half a meter which is consistent with the schematic in slide number 16 Here is the easterly wind And above that is the quasi-hydrostatic height And the hydrostatic approximation overestimate the height Back to slide number 18 And let’s compare the height bias to the height variability We used the ERA-interim data in the tropical region and filtered for wavenumber 1 to 10 And the horizontal standard deviation is roughly 10 m at 500 hPa So the height bias is on the order of 5% of height variability Although such a height bias may be smaller than random error in height estimation with rawinsonde it points out a source of systemic bias in model initial conditions Then, slide number 19 Knowing that both the vertical nontraditional Coriolis term and the buoyancy are contributors to thickness between pressure levels Here shows the zonal vertical distribution of these two contributors to the variability of thickness associated with the MJO The vertical nontraditional Coriolis terms are contoured And the buoyancy is shaded These are calculated from the ERA-interim data and are regressed upon MJO-filtered tropical rainfall at 90 degree East which is marked with the green arrow here On the west side of the green arrow which is in the rear flank of the active MJO there is a region dominated by upward nontraditional Coriolis term with weak buoyancy signals On the east side which is in the forward flank of the active MJO there is a region dominated by positive buoyancy slightly offset by downward nontraditional Coriolis term In terms of the maxima of absolute values the vertical nontraditional Coriolis term is roughly 5% of the buoyancy OK. Slide number 20 We’ve talked about how the nontraditional Coriolis terms affect hypsometric relation Note that I used surface pressure as a lower bound for upward integration because I used rawinsonde data For model data assimilation, however Such a reliable reference level does not exist So a thickness bias can contribute to height bias above or below Also note that the random error in rawinsonde observation may be larger than the systemic bias here However, even if the winds, pressure, temperature, and moisture contents were all well perfectly observed the assimulated height would still be substantially biased if the nontraditional Coriolis terms are omitted With that in mind Let’s move on to the equatorial wave propagation mechanism Slide number 21 I’ll start with the structure of an analytical solution of a wave Here shows the zonal vertical distribution of its mass stream function along the equator In the tropical atmosphere adiabatic warming or cooling is usually partially offset by diabetic cooling or heating which reduces the buoyancy generated Let’s first consider an extreme case where adiabatic and diabetic effects completely offset each other leaving no buoyancy Under such circumstances Will these circulations propagate? The normal answer is no because no buoyancy means no gravity waves However, we can see another wave propagation mechanism if we take both the nontraditional Coriolis terms and the vertically decreasing density into account The shading here denotes meridional planetary vorticity divided by density or 2 Omega / rho The compressional beta-effect is a vertical advection of the shaded quantity And it transmits these circulations eastward For example, the circulation in the middle is associated with positive meridional vorticity which points into the screen And the downward motion on the east side yields positive advection of the shaded quantity which increases the meridional vorticity on the east side The opposite is true for the west side So the circulation propagates to the east Then, slide number 22 Compressional beta-effect is first described in a paper by Verhoeven and Stellmach who derived the compressional Rossby wave solutions on an x-z plane with a mistake in their derivation This study corrects the mistake and further expand the spatial domain to a 3-dimensional equatorial beta-plane Theories about waves on an equatorial beta plane have a long history Matsuno derived the solutions with the shallow water model In Holton and Hakim’s textbook we can find the solutions with an anelastic model but without the nontraditional Coriolis terms Fruman as well as Roundy and Janiga derived the solutions with the nontraditional Coriolis terms but with a Boussinesq model None of these studies found the compressional beta-effect because it needs both the nontraditional Coriolis terms and the vertically decreasing density And the present study accounts for both of these ingredients In addition, in my derivation I put a factor alpha in the adiabatic term in the thermal dynamic equation alpha = 1 means dry adiabatic alpha between 0 and 1 means partial offset of adiabatic effects by diabetic effects where buoyancy frequency is effectively reduced alpha = 0 means complete offset or effectively neutral Then, slide number 23 Here shows the dispersion relations on a wavenumber-frequency space for a strongly stable atmosphere where the effective buoyancy frequency is far larger than the meridional planetary vorticity Curves on the right side denote eastward propagating waves and the left side denotes westward propagation For the same wavenumber a higher frequency means faster phase speed In the strongly stable case my solution is like the Matsuno’s solution and you cannot distinguish the black and red curves where black denotes with the nontraditional Coriolis terms and red without Now watch and listen to this video (Sound of piano played down the chromatic scale for 10 octaves) The sound frequency played here is proportional to the effective buoyancy frequency used to plot each frame of the animation Please pay attention to the frequency axis and the separation of black and red curves Let’s watch again (Sound of piano played down the chromatic scale for 10 octaves) With the effective buoyancy frequency decreasing the phase speed also decreases As you can see the frequency axis shrinks And the solutions with and without the nontraditional Coriolis term separate farther while the frequency - the effective buoyancy frequency - is decreasing In the last frame it is effectively neutral and the solutions without the nontraditional Coriolis terms no longer exists But we still find solutions with the nontraditional Coriolis terms And this is the compressional Rossby wave solution which propagates eastward at a phase speed of ¹⁄₄ m/s Then, slide number 24 This is a new figure created yesterday The compressional beta-effect can serve as a contributor to the eastward propagation of MJO Here shows the zonal vertical distribution of the compressional beta- effect which is shaded and the local tendency of meridional vorticity contoured These are calculated from the ERA-interim data and are regressed upon MJO-filtered tropical rainfall at 90 degree East which is marked with the green arrow In the mid-upper troposphere in the MJO-active region over the arrow and around - this region - the meridional vorticity is decreasing as you see the dashed contours Or in other words, the vertical wind shear is changing toward easterly Meanwhile, in the same region, the upward motion in the active-MJO yields negative advection of planetary vorticity divided by density In other words, there is a negative compressional beta-effect In terms of the minimum negative values the compressional beta-effect contributes 11% of the meridional vorticity tendency associated with the MJO In summary, slide number 25 In linear models, the nontraditional Coriolis terms affect the flow response to heat sources by roughly 10% and the hypsometric relation by roughly 5% and the equatorial wave phase speed by roughly ¹⁄₄ m/s or the meridional vorticity tendency associated with MJO by roughly 10% Unlike many numerical errors in the models these effects don’t start small and grow large with the time integration Instead, omitting these effects biases the initial pressure height relation and biases the zonal wind and wave propagation every time step These biases may grow even larger if nonlinear processes are considered which is left for future studies These results encourage restoring the nontraditional Coriolis terms into the models for more-accurate simulations for MJO, CCEWs, and ITCZ-forced flow This restoration is not going to resolve all the problems and the importance of improving the representation of physical processes is not downgraded However, the restoration of the nontraditional Coriolis term should be among the top priorities of model development because all other parts of the development depend on the choice of governing equations And to validate the implementation of the nontraditional Coriolis terms the compressional Rossby wave test can be used as a benchmark Slide number 26 This test aims to push the dry-dynamical parameters to a limit that is relevant to moist physics but difficult to test without modeling the moist physics Accordingly, I used three special treatments equatorial f-plane, fast planetary rotation, and barotropic gas I included the nontraditional Coriolis terms into the MPAS atmospheric dynamical core and used the upgraded dynamical core to simulate the compressional Rossby waves And the animation you see here shows the results The numerical solution in green contours overlaps the analytical solution in black contours So you can’t really distinguish the two types of contours which validates the implementation of the nontraditional Coriolis terms For the future, slide number 27 I have a proposal titled Effects of the nontraditional Coriolis terms in atmospheric models It aims to explore possible nonlinear effects and possible linkage to biases in ITCZ configuration and MJO propagation I plan to assist the inclusion of the nontraditional Coriolis terms into the NOAA FV3 model It may lead the modeling community to a new era where people build their models upon a sufficiently accurate governing equation With this proposal, I was awarded the NOAA Climate and Global Change postdoctoral fellowship But I declined this award because it doesn’t allow bringing the fellowship to an institution other than the proposed one So the proposed study is to be done at UC Davis Finally, slide number 28, acknowledgments This study would not have been done without my advisor Paul Roundy I first knew Paul by an email correspondence in 2016 That was before I attended the Conference on Hurricane and Tropical Meteorology in San Juan, Puerto Rico Paul, as a chair of that conference, sent the attendees an email about something that I don’t remember But I still remember his email signature here See that creature climbing on the business card With that first impression, then two years ago I asked Paul to be my advisor after I googled the term “nontraditional Coriolis terms” and found Paul’s paper And this turns out to be a great decision Paul cares for me, supports me, and guided me Paul is also extraordinary accessible His office is usually open for walk-in advice And he usually responds quickly to emails So, thank you, Paul Slide number 29 Many thanks to my other committee members Brian Rose, Rob Fovell, and Bill Skamarock Their comments improved the quality of this dissertation Bill also provided a tutorial that makes the development of the MPAS possible in this study Many thanks to my audiences First thanks to those attended my defense rehearsal Ahmed, Ajay, Hsiao-chun, Minghao, and Fangze Also, thanks to those attended my (presentation in) Prospectus Defense, Climate Group Meetings, Northeast Tropical Workshop, AMS Annual Meeting, and AGU Fall Meetings Moreover, thanks to those attended my invited talks at UC Davis, MIT, NCAR, and Taiwanese institutions including NTU, Academia Sinica, and Central Weather Bureau The reactions of my audiences helped me improving my presentation Many thanks to DAES staffs, Chaina, Annie, Barb, Kevin, and Ash for making all those administrative and technical issues so easy for me Slide number 30 Many thanks to my former PhD adviser Wei-Chyung Wang for bringing me in and caring for me Many thanks to my masters advisor at NTU, Chien-Ming Wu and Hung-Chi Kuo for their rigorous training in scientific writing and atmospheric dynamics Many thanks to Marat Khairoutdinov for his inspiring talk on March 26, two years ago Many thanks to all DAES and ASRC faculties and students for making such a friendly research environment Many thanks to my teammates in our soccer team, Snow Squall Squad Many thanks to my wife and my children Slide number 31 Many thanks to those who funded me during my three-year study ASRC offered me the graduate fellowship in my first year DAES has been offering me the teaching assistantship in my second and third year Taiwanese Ministry of Education has been offering me the Government Scholarship to Study Abroad in my third year Paul’s grant from the NSF funded my summer wages and trips to conferences My parents and parents-in-law partially funded some of my needs like my car and trips to Taiwan Next slides If you are interested in using the linear forced-dissipative model or the compressional Rossby wave test please find the Matlab scripts or the Fortran module below And that’s all, thank you .