Wind-forced solutions:
Equatorial ocean A short course on: Modeling IO processes and phenomena INCOIS Hyderabad, India November 1627, 2015 References 1) (HIG Notes) McCreary, J.P., 1980: Modeling wind-driven oceancirculation. JIMAR , HIG 80-3, Univ. of Hawaii, Honolulu,64 pp. 2) EquatorialNotes.pdf 3) Shankar_notes/equatorial_ocean.pdf 4) McCreary, J.P., 1981b: A linear stratified ocean model of theEquatorial Undercurrent. Phil. Trans. Roy. Soc. Lond., 298A, 603 635. 5) McCreary, J.P., 1985: Modeling equatorial ocean circulation. Ann.Rev. Fluid Mech., 17, 359409. Equatorial phenomena: quasi-steady currents
An Atlantic section along 23W (https://www.sfb754.de/l-atalante). A deeper Pacific section along 159W, showing deep equatorial jets (Ascani et al., 2010; JPO). Large-scale coastal phenomena forced by steady y include: i) alongshore coastal currents, namely, a surface jet in the direction of the wind and an opposite-flowing CUC; and onshore/offshore flow, which for upwelling-favorable winds consists of onshore flow at depth, coastal upwelling, offshore Ekman drift. Coastal phenomena forced by variable y are the radiation of coastally trapped waves along the coast and of Rossby waves offshore. Shelves also impact coastal phenomena, for example, allowing shelf waves and inhibiting the offshore propagation of Rossby waves. Coastal flows are also driven by Q.For example, when Q cools T(xe,y,0) poleward, Kelvin-wave adjustments cause the surface layer to thicken poleward roughly quadratically (Sumata and Kubokawa, 2001). There is a remarkable set of near-equatorial quasi-steady currents in the Pacific Ocean. They include the surface westward SEC, and the subsurface, eastward, EUC and Tsuchiya Jets (SEUC & NEUC). Equatorial phenomena: quasi-steady currents
23W An Atlantic section along 23W (https://www.sfb754.de/l-atalante). A deeper Pacific section along 159W, showing deep equatorial jets (Ascani et al., 2010; JPO). Large-scale coastal phenomena forced by steady y include: i) alongshore coastal currents, namely, a surface jet in the direction of the wind and an opposite-flowing CUC; and onshore/offshore flow, which for upwelling-favorable winds consists of onshore flow at depth, coastal upwelling, offshore Ekman drift. Coastal phenomena forced by variable y are the radiation of coastally trapped waves along the coast and of Rossby waves offshore. Shelves also impact coastal phenomena, for example, allowing shelf waves and inhibiting the offshore propagation of Rossby waves. Coastal flows are also driven by Q.For example, when Q cools T(xe,y,0) poleward, Kelvin-wave adjustments cause the surface layer to thicken poleward roughly quadratically (Sumata and Kubokawa, 2001). and a similar set exists in the Atlantic Ocean. Equatorial phenomena: quasi-steady currents
23W An Atlantic section along 23W (https://www.sfb754.de/l-atalante). A deeper Pacific section along 159W, showing deep equatorial jets (Ascani et al., 2010; JPO). 159W Further, there is a remarkable set of deeper currents as well. Nearly steady currents like those in the Pacific and Atlantic dont exist in the Indian Ocean because the monsoon winds are highly variable. Equatorial phenomena: El Nino
Sea level movie A similar ocean transition occurs during IOD events, with an IOD analogous to La Nina. La Nina in the Pacific is analogous to a positive IOD event in the Indian Ocean. Equatorial phenomena: Indian Ocean
Sea level movie In the Indian Ocean, steady equatorial currents are weak because there is almost no steady component to the wind forcing. As a result, the most prominent feature of the equatorial currents are the semiannual eastward flows (Wyrtki Jets). Questions What forcing mechanisms drive equatorial currents?
zonal and meridional wind stress What are equatorial waves? equatorial gravity, Rossby, and Kelvin waves; mixed Rossby/gravity (Yanai) wave How do they differ from midlatitude waves? dynamically very similar; extra Yanai wave; discreteness What are the key differences between 2-d and 3-d theoriesof equatorial circulation? Yoshida Jet; establishment of px to balance x How do equatorial waves reflect from basin boundaries? Kelvin- and Rossby-wave reflections; critical latitude Some questions and possible answers. Introduction Equatorial waves Solutions for switched-on winds
Solutions for periodic winds Equatorial waves Equatorial-ocean equations
Equations for the un, vn, and pn for a single baroclinic mode are (1) Because f vanishes at the equator, no terms can be dropped that allow for mathematically simple solutions near the equator. A useful assumption, though, is to set f = y, known as the equatorial -plane approximation. As a result, one can look for solutions as expansions in Hermite functions. Equatorial gravity and Rossby waves
We look for free-wave solutions to (1) of the form, (y)exp(ikx it), without damping (A = 0), and, for convenience, we drop the subscript n.The resulting v equation is (2) (2) It is convenient to introduce the non-dimensional variable and to rewrite the v equation in terms of . The mathematical difficulty with obtaining a dispersion relation from (2) is that, because f varies so much near the equator, it is not possible to set () = exp(iy).Rather, (y) is the set of solutions (eigenfunctions) that satisfy where = 0, 1, 2, .They are referred to as Hermite functions. Reinserting subscript n, the length scale Rn = (on)1 = (/cn) is referred to as the equatorial Rossby radius of deformation.Note that it has a different value for each baroclinic mode n.Usually, its reported value is for the n = 1mode. With cn = 250 cm/s and = 2.28x1013 cm1s1, its value is R1 = 331 km. NOTE: I use in two different waves.In previous talks, it was the meridional wavenumber.Here, it is the index of a Hermite function and (as we shall see) an equatorial Rossby or gravity wave. Equatorial gravity and Rossby waves
The figure plots the first six Hermite functions ( = 05).The scaling factor, LR = Rn = (cn/), the equatorial Rossby radius of deformation for baroclinic mode n.(For n = 1, LRis roughly 331 km.)Note that the are less equatorially trapped (extend farther off the equator) as increases.Note also that they alternate between being symmetric and antisymmetric about the equator. For large , the Hermite functions resemble cosine or sine curves near the equator.They begin to decay at latitudes higher than the turning latitude.So, the Hermite functions are equatorially trapped. Ascani (2002) Fedorov and Brown (2007) High-order Hermite functions are similar to sine waves, except that they cut off beyond a certain latitude, the turning latitude. They cut off when 2 becomes bigger than 2 +1.In that case the curvature of the response changes sign and the response becomes exponential rather than oscillatory. Equatorial gravity and Rossby waves
The solutions to (2) can be represented as expansions in Hermite functions (3) where v is a wave amplitude.Each term in expansion (3) is an individual equatorial wave. Inserting term in (3) into (2) gives (2) which provides the dispersion relation for equatorial, Rossby and gravity waves. Equatorial gravity and Rossby waves
The dispersion relations for equatorial and midlatitude waves are very similar.They differ only in that = f/c varies continuously for midlatitude waves, whereas has discrete values for equatorial waves. For each > 1, there is a gravity wave (large ) and a Rossby wave (small ).The plot shows waves for = 1, 2, and 3. /o k/o 1 3 For = 0, there is a new type of wave, the mixed Rossby-gravity (Yanai) wave, which joins the Rossby-wave (gravity-wave) wave curves for large negative (positive)values of k. Matsuno first published and discussed this famous dispersion relation. Mixed Rossby-gravity (Yanai) wave
The curious form of the Yanai-wave dispersion curve happens because it factors into two parts when = 0.We have Mixed Rossby-gravity (Yanai) wave
The curious form of the Yanai-wave dispersion curve happens because it factors into two parts when = 0.We have The second factor describes a wave that travels westward at the speed of a Kelvin wave.It can be shown that this wave blows up at , and so it must be discarded. The single dispersion relation for the Yanai wave is then For small and large values of , the relation simplifies to, the same properties for Rossby and gravity waves, respectively. Equatorial gravity and Rossby waves
The v, u, and p fields for equatorially trapped Rossby and gravity waves are whereV is a constant amplitude, and j = 1 (2) corresponds to the (+) sign. Equatorial gravity and Rossby waves
k/o 1 3 The u, v, and p fields for a Yanai wave when cn = 250 cm/s and P = 30 days.For this P, /o = .36 and = 7.3. Courtesy of Francois Ascani Equatorial gravity and Rossby waves
k/o 1 3 The u, v, and p fields for a Yanai wave when cn = 250 cm/s and P = 360 days.For this P, /o = .03 and = 0.64. Courtesy of Francois Ascani Equatorial gravity and Rossby waves
k/o 1 3 The u, v & p fields for an = 1 Rossby wave when cn = 250 cm/s and P = 360 days.For this P, /o = .03 and = 240. Courtesy of Francois Ascani Equatorial gravity and Rossby waves
k/o 1 3 The u, v, and p fields for an = 2 Rossby wave when cn = 250 cm/s and P = 360 days.For this P, /o = .03 and = 140. Courtesy of Francois Ascani Equatorial Kelvin wave
The equatorial Kelvin wave has v = 0, and so was missed in the preceding solutions.To find it, set v = A = 0 in (1), and look for a free-wave solution of the form (y)exp(ikx it). With these restrictions, equations (1) reduce to The first and third equations imply and the second then gives (4) Equatorial Kelvin wave
The solution to (4) is The solution that grows exponentially in y, which corresponds to the root, k = /c, is physically unrealistic in an unbounded basin and must be discarded.Therefore, the only possible wave is (5) The y-structure of the Kelvin wave is o(y), the lowest-order Hermite function. which describes the structure and dispersion relation for the equatorial Kelvin wave.In (5), I have used the property that and redefined the arbitrary constant amplitude to be Po = P'o. Theoretical equatorial waves
To summarize, for each > 0, there is a gravity wave (large ) and a Rossby wave (small ).The plot indicates the waves only for = 1, 2, and 3. /o k/o 1 3 There is a mixed-Rossby-gravity wave for = 0, that behaves like a Rossby (gravity) wave for k positive (negative). There is an equatorial Kelvin wave. A lot of mathematics led to this set of dispersion curves.Do any of these waves actually exist?! Observed equatorial waves
Tom Farrar (2010) The first equatorially trapped waves to be discovered were gravity-wave resonances with periods of O(10 days) (Wunsch and Gill, 1976; Deep-Sea Res.).There are no publications that explore the possibility of Rossby-wave resonances. /o k/o 1 3 The equatorial Kelvin wave was discovered after it was predicted (Knox and Halpern, 1982, JMR). The mixed Rossby-gravity (Yanai) wave was first observed in the atmosphere by Yanai.In the ocean, it was (probably) first detected in the Indian Ocean by Reverdin and Luyten (1986) using altimeter data. Movies E Who first detected an equatorial Rossby wave? Solutions for switched-on winds x-independent (2-d) Yoshida Jet
Kozo Yoshida wrote down the first solution for an x-independent (2-d) equatorial current driven by zonal winds.The (more complete) theoretical solution developed somewhat later (Dennis Moore) has come to be called the Yoshida Jet (Jim OBrien). The basic dynamics of the Yoshida Jet can be understood from the zonal-momentum equation. Neglecting the pressure-gradient and mixing terms in the zonal momentum equation gives Offshore, Ekman balance (fvn = x/Hn) holds, whereas at the equator un continues to accelerate (unt = x/Hn).The switch from one dynamical regime to the other occurs at y on = (/cn) . Bounded (3-d) Yoshida Jet
Zonal flows along the equator (Yoshida Jets) in reality and models dont continue to accelerate.Why not? Because in the real world either the wind forcing or the ocean basin is zonally bounded, which introduces x-dependence into the solution.(An exception is the Southern Ocean, but we will not consider that case here.) For convenience, we can still drop the mixing terms in the zonal momentum equation, and at the equator the Coriolis term vanishes.The boundaries, however, introduce x-dependence so we cannot neglect the pnx term In this case, the system can stop accelerating by adjusting to a state where the pressure gradient balances the wind. It does so by radiating equatorial Kelvin and Rossby waves. Bounded (3-d) Yoshida Jet
d (1 month) d (6 months) Equatorial jet Kelvin wave Rossby wave Coastal KW Rossby wave Kelvin wave 1) These process occurs throughout the year in the Indian Ocean, in response to the semiannual oscillation of the equatorial winds. What happens when the basin boundaries are included? Suppose that the ocean basin is unbounded but the wind is bounded, a patch of zonal wind. In response to forcing by a patch of easterly wind, an acceleratingYoshida Jet initially develops in the forcing region.Subsequently, KWs and RWs radiate from the forcing region.They generate a steady, eastward, equatorialcurrent both east and west of the forcing region: the bounded YJ. Eastern-boundary reflections
For low frequencies, the incoming Kelvin wave reflects as a packet of Rossby waves (Moore, 1968).with the waves corresponding to larger values propagating offshore more slowly. 1 3 5 7 Movie G1 The zonal current of the Kelvin wave divides at the Rossby-wave front to flow along the edges of the wave packet. Adjustment to steady state
d (1 month) In response to forcing by a patch of wind in the interior ocean, KWs reflect from the eastern boundary as a packet of RWs creating a characteristic wedge-shaped pattern.In addition, wind-generated RWs reflect from the western boundary to return to the interior ocean. Rossby wave Kelvin wave d (6 months) Coastal KW Rossby wave Kelvin wave Equatorial jet d (1 year) Reflected Rossby waves After multiple reflections, the solution eventually adjusts to a state of Sverdrup balance. In response to an annual or semiannual oscillation of the equatorial winds, these adjustments continue to happen throughout the year in the Indian Ocean. Rossby wave d (5 years) Reflected Rossby waves Near Sverdrup flow Eastern-boundary reflections
Remarkably, the characteristic wedge shape and westward propagation is visible in satellite data.The figure shows global maps of filtered sea level from TOPEX/Poseidon on April 13 and July 31, It shows a Rossby-wave packet generated by the reflection of an equatorialKelvin wave forced by intraseasonal winds in the western ocean. (After Chelton and Schlax, 1996.) Movies F In Movies F, for zonal winds Kelvin and Rossby waves radiate from the wind patch leaving behind a Sverdrup balanced flow + a bounded Yoshida Jet. For meridional winds, Yanai and Rossby waves radiate from the wind path to establish a Sverdrup circulation.There is no bounded Yoshida Jet because meridional winds cannot generate one (wrong symmetry). Fedorov and Brown, 2007 Multi-baroclinic mode adjustment
How does the LCS model adjust when many baroclinic modes are included? d (1 month) The plot shows the n = 1 responsewithout damping. It also illustrates the n > 1 responses, except that thecurrents are narrower in y (Rn < R1) and the wave speeds are smaller (cn < c1). Rossby wave Kelvin wave d (6 months) Coastal KW Rossby wave Kelvin wave With damping, the n > 1 responses are increasingly damped since = A/cn2.In that case, waves that radiate from the forcing region are weakened for larger n.For sufficiently large n, then, the response is confined to the forcing region. Equatorial jet d (1 year) Reflected Rossby waves See McCreary (1981) for a detailed description of how the structure changes with n. Rossby wave d (5 years) Reflected Rossby waves Near Sverdrup flow Multi-baroclinic mode adjustment
When the LCS model includes damping (vertical mixing), a realistic steady flow field is produced near the equator. EUC McCreary (1981) Movies I1a, I1b & I1c Solutions for periodic winds Evanescent waves /o k/o
As for the coastal model, there are two wavenumbers, k1,2, associatedwith each value.The wavenumbers k1 (k2) describe waves with eastward (westward) group velocity or decay. /o k/o 1 3 Also as for the coastal model, the wavenumbers are real for small (Rossby waves) and become complex as increases. Eventually, they become real again for even larger (gravity waves). The region of complex roots for = 1 waves is indicated by the shading.Such waves exist only along boundaries, where they superpose to generate -planecoastal Kelvin waves. Movies G2 and G3 Critical frequencies /o k/o
Free Rossby waves exist at frequencies only below the shaded region. Consider how the ocean responds to oscillatory winds at different frequencies.At frequency 1, the wind can only excite Kelvin and Yanai waves.At frequency 2, it can also excite = 1 RWs.At frequency 3,it can also excite = 1, 2, and 3 RWs. /o k/o 1 3 Movies G2 and G3 Vertical propagation Recall that the vertical structure ofwaves in the LCS model satisfy Rather than to look for solutions as expansions in vertical modes, n(z), another way of studying solutions to the LCS model is to look for approximate solutions of the form, under the restriction that the background stratification, Nb(z) varies slowly with respect to the vertical wavelength of the wave, m(z) (the WKB approximation). In that case, Since cn > 0, the replacement in the last equation already assumes that m > 0.if m < 0, then the replacement should be cn = Nb/m.So, the logic of these slides is a bit wrong, as they dont correctly state where the assumption that m > 0 is made.[NOTE: To fix this problem, I replaced m with |m| in the last equation on this page and the top one on the next page.] and cn can be replaced by Vertical propagation (KW beams)
With this change, the dispersion relation for equatorial Kelvin waves is Group theory states that a packet of Kelvin waves (that is, a superposition of several waves associated with different k and m values) propagates at the group velocity Thus, the energy of the packet propagates to the east with the slope Along a northern coast, = cnk, and the signs of all the propagations are reversed. Similar results hold along north-south oriented boundaries, with the complication that Kelvin waves associated with a particular baroclinic mode dont exist equatorward of the critical latitude. The movies show Kelvin beams at several different frequencies, radiating downward along beam paths. So, if phase propagates upwards (m > 0), energy propagates downwards, and vice versa. Vertical propagation (YW beams)
The dispersion relation for Yanai waves becomes Group theory states that a packet of Yanai waves (that is, a superposition of several waves associated with different k and m values) propagates at the group velocity Along a northern coast, = cnk, and the signs of all the propagations are reversed. Similar results hold along north-south oriented boundaries, with the complication that Kelvin waves associated with a particular baroclinic mode dont exist equatorward of the critical latitude. The movies show Kelvin beams at several different frequencies, radiating downward along beam paths. Thus, the energy of the packet propagates to the east with the slope the same slope as for Kelvin waves! Vertical propagation (long-wavelength RWs)
For the RW dispersion curves, as tends to zero so does k. So, in the low-frequency limit the RW disp. curves are non-dispersive. This limit is known as the long-wavelength approximation. /o k/o 1 3 In this limit, RWs propagate vertically with a slope with a steeper slope, and in the opposite direction from, KW and YWs. Single baroclinic mode response
d (1 month) In response to forcing by a patch of switched-on, easterly winds, Kelvin and Rossby waves radiate from the forcing region, reflect from basin boundaries, and eventually adjust the system to a state of Sverdrup balance. Rossby wave Kelvin wave d (6 months) If the wind oscillates, say, at the annual, semiannual, or intraseasonal periods, waves are continuously generated. Equatorial KWs and RWs continuously radiate from the forcing region.Coastal KWs radiate around the perimeter of the basin, and eastern-boundary RWs radiate into the interior ocean. Equatorial jet d (1 year) Reflected Rossby-wave packet In response to an annual or semiannual oscillation of the equatorial winds, these adjustments continue to happen throughout the year in the Indian Ocean. d (5 years) Sverdrup flow Multi-baroclinic mode response
Recall that when the LCS model includes vertical mixing (damping), a realistic steady flow field is produced near the equator.What happens if the wind oscillates, say, at annual, semiannual, or intraseasonal periods? EUC The equatorial Indian Ocean is dominated by vertically propagating equatorially trapped waves.Indeed, there are virtually no steady flows at all! McCreary (1981) Without damping, waves radiate from the forcing region along beams that extend into the deep ocean and exhibit upward phase propagation.Yanai and Kelvin beams extend downward and eastward, and Rossby waves extend downward and westward. With damping, the beams weaken away from the forcing region. Tropical instability waves
Legeckis (1977, Science) first reported the presence of TIWs in the eastern, tropical Pacific.TIWs were soon shown to have a large impact on the momentum and heat fluxes in the region.Philander (1976, 1978, JGR) argued that TIWs were caused by barotropic instability.Yu et al. (1992, Prog. Oceanogr.) later suggested that an instability of the temperature front was involved.Luther and Johnson (1990) suggested that there was more than one type of TIWs. 1) Note that clouds seem to be following the TIW front in the northern hemisphere.Modeling work in the past decade has explored the impact of TIWs on the atmosphere, using both AGCMs and CGCMs. Similar TIWs were soon observed in the Atlantic Ocean.Their dynamics are essentially the same as for the Pacific TIWs. Tropical instability waves
Cox, M.D., 1980: Generation and propagation of 30-day waves in a numerical model of the Pacific. J. Phys. Oceanogr., 10, 11681186. The above solution assumes that Nb is constant.When Nb weakens with depth, as it does in the real ocean, ray paths slope more steeply with depth. Michael Cox (1980) reported a Yanai-wave beam forced by surface TIWs in his OGCM solution.Ascani & coworkers (2009) explored the idea that deep equatorial currents are caused by an instability of the Yanai-wave beam generated by TIWs.To simulate the effect of TIWs, they forced their OGCM by a wind stress with the wavelength (~1000 km) and period (~30 days) of a typical TIW, generating the Yanai-wave beam shown above. Tropical instability waves
Harvey and Patzert (1976) likely detected the off-equatorial u field of the TIW-driven Yanai beam on the ocean bottom east of the Galapagos.(Sadly the mooring on the other side of the equator failed.) Upward phase propagation in the EEIO
Masumoto et al. (2005) The u field (b & d) shows a strong semiannual cycle. Above 200 m, the phase of upropagates upwards, indicating that it is remotely forced (wave) signal! There is a difference in the signals visible in u and v, with the latter exhibiting strength in the 1020-day band. Note the upward phase propagation of the semiannual signal, a property that the LCS model can easily simulate. Movies J Bounded (3-d) Yoshida Jet
If the pressure-gradient term pnx is then included, the flow field has both a realistic amplitude and structure. If only the damping term (A/cn2)un is included in the zonal momentum equation, the flow stops accelerating, but it is unrealistically fast (the unit is km/s) and extends to the bottom. Zonal flows along the equator dont continue to accelerate in reality or models.Why not? Eastern-boundary reflections (Moores chain rule)
Suppose the ocean is forced by a patch of oscillating zonal wind confined to the interior ocean.It generates an equatorial Kelvin wave, that radiates to the eastern boundary of the basin. There can be no zonal flow through the boundary.How does the system adjust to prevent this flow? Dennis Moore showed that a packet of equatorial waves with the zonal velocity field, (7) are generated at the eastern boundary.In (7), the wavenumbers k1 correspond to waves with westward group velocity or decay. Eastern-boundary reflections (Moores chain rule)
For convenience, let the eastern boundary be located at x = 0.The uK field there is and it must be cancelled by To eliminate uK, we use the u1 wave and set B1 = Uo.With this choice the 0 terms are cancelled, but a 2 term is created.We use the u3 wave to cancel this term, and so on. In general, once B is known then the recursion relation for Moores famous chain rule. Long-wavelength approximation
Equations for the un, vn, and pn for a single baroclinic mode are In the equatorial region, there are no simplifications that allow for mathematically simple solutions (i.e., that allow y-derivatives to be dropped. One useful simplification (analogous to the coastal one) is to adopt the long-wavelength approximation, which restricts the zonal flow to be in geostrophic balance.For convenience, also drop horizontal mixing from the x-momentum equation (but that is not necessary).