Diagnosis of a Quasi-Geostrophic 2-Layer Model Aaron Adams, David Zermeño, Eunsil Jung, Hosmay Lopez, Ronald Gordon, Ting-Chi Wu Introduction For this project we use a simple two layer model, which is more suitable for the ocean, to demonstrate the effect of baroclinicity. This problem was first studied by Phillips (1951). In the two layer system, the ocean or atmosphere are represented by two discrete layers, therefore only the barotropic and first baroclinic modes will be present in the system. To simplify the problem even more, a constant vertical shear is used in each layer. This still allow us to analyze baroclinic instabilities. In the first part of the study, eddy energy evolution and its role in momentum and heat transports are studied using the two layer QG channel model. The model is run with different background flows from 6 to -3 cm/s. Results with U 0 =-3cm/s, U 0 =3cm/s and U 0 =6cm/s are presented here. The effect of bottom friction on mean zonal flow is analyzed by running the two layer model using the default conditions of the model. Only the bottom friction is changed and evaluated using frictional values of 0, 2x10-7 /s, 4x10-7 /s, and 6x10-7 /s. Intermediate values were evaluated and found to be within the trend observed and were subsequently omitted. It is found that changes in bottom friction result in changes in eddy formation. We learned the two-layer Phillips model with equal thickness in class. In another part of this study, we look at how varying the thickness of the lower layer affects the dynamics of the flow. The upper layer thickness is fixed at 1km for all the five experiments, and the lower layer thickness is varied between 1km and 5km. Thickness of lower layer for 1km and 2km are theoretically beyond their critical shear and becomes unstable very fast. Since we are interested in unequal thickness effects on the two-layer QG model, we focus on the other three cases and not further investigate the 1km and 2km experiment. In another part of this study, eddy development is investigated within the framework of the theoretical Phillip's equal thickness, two-layer quasi-geostrophic model with a varying Mean Zonal Background (MZB) flow. It is found that for eastward values, eddies are stronger in the top layer while for westward flows the opposite is true. Additionally, it is discovered that when the absolute value of the MZB is near zero, a distinct 180 degree phase shift exists between activity in the top and bottom layer. This will be further discussed. In the next section of the study, the two layer quasi-geostrophic model is initialized with three distinct initial perturbations. The goal is to validate the importance of vertical shear and wavelength for the development and growth of baroclinic instabilities. All disturbances have the same initial amplitude of -2cm for the top layer and -0.5cm for the bottom layer, but different shear and wavenumber. Only one perturbation will be under favorable conditions for development. 1
Finally, the unstable simulations made in the previous experiment are further analyzed by computing the Eliassen-Palm flux vector and its divergence. A description of the energy supplied from the background flow to develop Rossby waves from such initial perturbation is followed. Part A: Eddy Energy Evolution and its Role in Meridional Momentum and Heat Transports in Different Background Winds. 1. Introduction and Experiment Eddy energy evolution and its role in momentum and heat transports are studies in the two layer QG channel model. QG-model has run with different background winds from 6 to -6 cm/s. Results with U0=-3cm/s, U0=3cm/s and U=6cm/s are presented here. 2.1 Energy Evolution To investigate the evolution/growth of eddies, zonal kinetic energy (hereafter KZ) and eddy kinetic energy (hereafter KE) and conversion between them (hereafter CK) are calculated (James, 1994). (1); (2); (3) Where < > and * indicate volume integration and departure from zonal average [ ], respectively. To simplify, the mean value of energy is used in this calculation instead of volume integrated value. Time evolution of KE is shown in Figure 1. Solid (dotted) lines indicate KE of layer 1(2); black line for 3cm/s, red for 6cm/s and blue for -3cm/s. Fig. 1: Time series of eddy kinetic energy (KE) for U 0 =3cm/s (red), U 0 =-3cm/s (blue) and U 0 =6cm/s (black), respectively. Solid line corresponds to the layer 1 (layer 2 for dotted line). Figure 1 shows that KE with westerly background wind (U 0 =3cm/s and 6cm/s) has greater energy in layer 1 (upper layer) than that of layer 2, while easterly flow (U 0 =-3cm/s) is more 2
energetic in layer 2 (bottom layer). Eddies in the easterly background wind (U 0 =-3cm/s) grow faster than that in the westerly background wind (U 0 =3cm/s) and have more kinetic energy if eddies in the same magnitude of background wind are compared. Eddy growth rate can be estimated from time series of KE. For the easterly background wind (U 0 =-3cm/s), eddies in layer 2 grow fast and reach a maximum growth rate ahead of layer 1. However, it is not clear for the case of U 0 =6cm/s. Regarding U 0 =3cm/s, KE decreases with time. It shows some bumps but does not grow like the others, indicating it does not have critical shear. Evolution of eddy kinetic energy is illustrated in Figure 2. An x-axis (denoted by Y in Fig. 2) indicates a latitudinal distance. Fig. 2: Zonally averaged eddy kinetic energy at a different time sequence (for U 0 =6cm/s). Figure 2 shows that eddy kinetic energy starts to increase from both the boundaries and it propagates into the interior suggesting it results from no-slip boundary condition. After reaching the maximum energy level in the interior of the domain, KE slowly decrease without any remarkable pattern changes. Time evolution of KE, KZ and energy conversion from KE to KZ are shown in Figure 3. 3
Fig. 3: Time series of two-layer averaged KE, KZ and CK for u0=6cm/s. Two-layer averaged KE is calculated by and the same manners are applied to obtain KZ and CK. Figure 3 shows that maximum growths of KE and KZ occur at slightly different times. Rapid growth of CK is observed between KE and KZ peaks. In case of U 0 =6cm/s, maximum growth rate occurs at around 230~260 days. Eddy components of U field (shown as U*) during this period are shown in Figure 4. Fig. 4: U* field at day 254 and day 264 for U 0 =6cm/s (Please notice that the color scale is different). Figure 4 shows that eddy strength (shown as U*) becomes the strongest (at day 253) when energy levels reach maximum, then it breaks down piece by piece showing more eddy structures (at day 264). Magnitudes of U and V are similar at this point (not shown here). Horizontal wind field at the time of CK reaches maximum is shown in Figure 5. Week stratification is clearly seen in the figure. (Not shown here but) U* field shows that upper level (layer1) has more eddy activity than lower layer (layer 2). Note that the lower layer has more eddy activity in the case of easterly background (U 0 =-3cm/s) flow. Fig. 5: U field at 242 day when eddy activity reaches at its maximum. 2.2 The Meridional Momentum and Heat Transports. In general, a quantity A can be expressed as and/or. Thus, momentum/heat transports were calculated by (Peixoto and Oort, 1992) For example, to calculate northward flux of sensible heat, we substitute A=v and B=c p T in the above equation. Bar here means time mean and [ ] indicates zonal mean. Total northward transport of heat (or momentum, A=u, B=v) on the left hand side of equation (4) is decomposed 4 (4)
into three terms: the transport by the (steady) mean meridional circulation, the stationary eddies, and the transient eddies, respectively. Transient eddies are useful for storm tracks. Figure 6 shows the transport by the (steady) mean meridional circulation (MMC), the stationary eddies, and the transient eddies for u0=6cm/s, u0=3cm/s, and u0=-3cm/s, respectively. Fig. 6: Transport by the (steady) mean meridional circulation (denoted as MMC), the stationary eddies, and the transient eddies for U 0 =6cm/s, U 0 =3cm/s, U 0 =-3cm/s, respectively. First of all, for the cases of U 0 =6cm/s and U 0 =-3cm/s, we can see that momentum transport by the transient eddies is much greater than that by the mean meridional circulation, indicating eddies are the primary mechanism maintaining the zonal mean zonal flow. As eddies get more energetic, the contribution of transient eddies on the total meridional transport gets larger (in this example, eddies in U 0 =6cm/s are more energetic than U 0 =-3cm/s, shown in Fig. 1). It also shows a different level of maximum intensity: maximum intensity is located in layer 1 (upper layer) when westerly background wind (U 0 =6 cm/s) is used (same as U 0 =3cm/s), while maximum transport occurs in layer 2 (bottom layer) when easterly U 0 (U 0 =-3cm/s) is simulated. An x-axis indicates a latitudinal distance from south to north and it displays basin scale of 1800km. For both cases (U 0 =6cm/s and U 0 =-3cm/s), 9-10 cells can be seen in Figure 6, implying the size of each cell is approximately 200km. These small cells represent eddies. Layer 1 and layer 2 are expressed in y-axis. Most of the cells (small circulations) for the case of U 0 =6cm/s are upright, indicating (equivalent) barotropic conditions. Small vertical tilts can be found when easterly background (U 0 =-3cm/s) is used. In the mean time, for the case of U 0 =3cm/s, the case that eddies didn t develop much, contributions of stationary eddies and transient eddies on the total transports are similar but at the same time the amount of transports are so small and can be negligible. It also shows only a large gyre circulation. Heat transports at the interface between two layers are shown in Figure 7. Temperature field is obtained from the difference of stream function (Ψ1-Ψ2) and the heat transport at the interface is calculated by. In case of U 0 =6cm/s, similar to the momentum transport, heat transport by the transient eddies is much greater than that by other 5
circulations. In addition, it shows positive mean meridional heat transports through the whole latitudes, implying it transports warm water (cold water) toward the poles (equator), acting to reduce the gradient of isopycnals by releasing potential energy to kinetic energy. In case of U 0 =3cm/s, which didn t contain much eddy activities, total meridional transport is very little. Nevertheless, contribution of stationary eddies on the total meridional transport is dominant and heat transports by the transient eddies are negative, but again, the amount is so small and can be negligible. Fig. 7: Meridional heat transport at the interface between two layers for U 0 =6cm/s (left) and U 0 =3cm/s (right). 3. Discussion and Results Eddy energy evolution and its role in momentum and heat transports are studied in the two layer QG channel model. Results from three different background winds are focused; U 0 =6cm/s (a case of vigorous eddy activity), U 0 =3cm/s (a case of subcritical shear), and U 0 =-3cm/s (easterly background wind, eddy active). Evolutions of eddy energy are investigated with zonal kinetic energy (KZ), eddy kinetic energy (KE) and energy conversion from KE to KZ. Results show that Eddies have higher kinetic energy in upper layer (layer 1) when westerly background winds are simulated, while eddies in layer 2 (bottom) show higher kinetic energy, grow fast and reach a maximum growth rate ahead of layer 1 when easterly background wind is used. Total northward transports of heat and momentum are examined to see the role of eddies in transports. In the case of U 0 =6cm/s and 6
U 0 =-3cm/s, momentum transport by the transient eddies is much greater than that by the mean meridional circulation and by the stationary eddies, indicating eddies (especially transient eddies) are the primary mechanism maintaining the zonal mean zonal flow. Small circulations caused by eddies (size of approximately 200km) are seen the circulation represented by the momentum transport. Eddies are shown as upright, indicating (equivalent) barotropic instability in this experiment. Heat transports by eddies show positive, implying transports warm water to the poles and cold water to the equator, reducing gradients of isopycnals by releasing APE to KE. For the case of U 0 =3cm/s, it doesn t play a role to transport momentum and/or heat. Part B: The Effects of Bottom Friction 1. Kinetic Energy Investigation In effort to evaluate the effect of bottom friction on the mean flow, total kinetic energy (KE total ) and eddy kinetic energy (KE eddy ) are first evaluated. KE total and KE eddy are calculated for 3000 days as follows: =12 2+ 2 ; =12 + 02+ 2 = ; = where u and v are area averaged wind values, is the stream function from model output, U 0 is background wind, and n indicates layer number. Values of U 0 are 6 cm/s in the top layer (n=1) and 0 cm/s in the bottom layer (n=2). Figure 1: Total kinetic energy (top) and eddy kinetic energy (bottom). Solid lines indicate top layer and dashed lines indicate bottom layer. Y- axis is displayed in log units for better visualization. A quick perusal of Figure 1 yields an initially counter-intuitive posture; smaller values of bottom friction have a larger magnitude of kinetic energy during growth, reach maximum growth rate at an earlier time step, and have a smaller magnitude of kinetic energy when kinetic energy values 7
become relatively stable than that of larger values of friction in the top layer. In the bottom layer, smaller values of bottom friction also have a larger magnitude of kinetic energy during growth and reach maximum growth earlier, but maintain the larger values of energy after reaching stable values. This counter-intuitive bearing can be explained by examining the transfer of energy between the layers and evaluating the wind fields as the kinetic energy is calculated directly from the wind field as described above. Figure 2: 3000 day time mean zonal wind contours (left) and time mean zonally averaged zonal wind with cubic fit (right) for each value of friction. 8
In order to understand the discrepancy between the top and bottom layers energy values, one must think about how the energy is transferring between the layers. When no friction is present, the flow in the bottom layer is allowed to move freely. When friction is introduced, the flow on the bottom is inhibited by the friction causing pressure changes and/or baroclinic instability. Because we are dealing with an incompressible fluid in the ocean, these changes result in a change in a height perturbation at the layer interface which introduces pressure changes which influence the magnitude of the wind fields. When a height perturbation is positive, the wind field must also increase in accordance with pressure changes. This, in turn, increases the time mean kinetic energy of the top layer. 2. Zonal Wind Field Figure 2 displays the 3000 day time averaged zonal wind field for each value of bottom friction. As friction increases, zonal velocity on the bottom layer decreases as one would expect. In the absence of friction, multiple zonal jets develop and are easily visible when looking at a plot of zonal wind vs. latitude. The displayed cubic fit indicates that the zonal mean is evenly distributed in the case of no friction. As friction is increased, eddy development in the upper layer increases, while the lower layer is still primarily comprised of zonal jets. The distribution of zonal mean zonal velocity changes with differing frictional values. A friction value of 2x10-7/s displays larger zonal mean values at the lower latitudes and negative or near zero values at the higher latitudes as seen in the cubic fit. As frictional values increase, the cubic fit migrates from positive values in the upper and lower latitudes with near zero in the middle latitudes to a fit with the most positive values at the upper latitudes. While it does appear the top layer zonal wind values are larger in the case of no friction, in a mean sense they are lower as indicated by the eddy kinetic energy displayed in figure 1. Figure 3: Relative vorticity for each value of friction. As the wind field changes with changes in pressure, relative vorticity (ζ) also changes as: 9
= As a consequence, the changes in the wind field due to different values of bottom friction change the relative vorticity as seen in Figure 3. In the absence of friction, the relative vorticity field closely mimics the zonal jets seen in the zonal velocity field. As friction increases, these zonal bands diminish in the top layer into small, non-uniformly distributed vortices, or eddies. 3. Results In the absence of bottom friction, energy transfer between layers is much less than the energy transfers using higher values of friction. As friction combats the zonal velocity, interface perturbations arise causing changes in the wind field due to pressure changes. The changes in the wind field cause changes in relative vorticity which in turn create more, less uniform, and smaller vortices, or eddies. The number of eddies that develop increases with increasing values of bottom friction. Part C: The Case of Unequal Layer Thickness In this QG two layer channel model, we will examine the eddy activities when different vertical stratification structures are introduced. Since vertical stratification in the two-layer model is determined by the Rossby radius of deformation (R d ), one can modify either the reduced gravity ( ) to increase/decrease the stratification intensity or the thicknesses of two layers (H 1 and H 2 ) to change the stratification structure. Here, we will apply them together so that we can extend Phillips theory into a more general case. 1. Model The quasi-geostrophic potential vorticity equation of the two layer Phillip s model: where We are interested in the unequal thicknesses of layers (H 1 H 2 ). So we expect to see different baroclinic Rossby deformation radii for the upper and lower layers (R d1, R d2 ). After assuming a wave-like solution, the imaginary part of the PV equation is: 10
For unstable modes to exist, the term in brackets need to equal zero. Therefore, we can have a nonzero imaginary part, ω i 0. This implies that and should have different signs. Since and, two scenarios that can lead to a change in sign of in the two layer model: (1) If shear (U 1 -U 2 ) > 0 then so (2) If shear (U 1 -U 2 ) < 0 then so 2. Experiments The meridional width of the channel L y = 1800 km. The background planetary vorticity gradient is, and the mid-channel (45 N) Coriolis parameter is. U 1 =6 cm/s and U 2 is zero, so the vertical shear in this experiment is 6 cm/s. Each case is run for 3000 days to see the affects of different thickness layers on the behavior of eddies. Here we designed five cases that change the H 2 thickness from 1 km to 5 km and keep H 1 the same. U 1 -U 2 > 0 Exp H 1 H 2 2 R d1 2 R d2 Critical Shear 1 1 km 1 km 1.23x10 9 m 1.23x10 9 m 0.025 m/s = 2.5 cm/s 2 1 km 2 km 0.93x10 9 m 1.86x10 9 m 0.037 m/s = 3.7 cm/s 3 1 km 3 km 0.84x10 9 m 2.52x10 9 m 0.050 m/s = 5.0 cm/s 4 1 km 4 km 0.77x10 9 m 3.09x10 9 m 0.062 m/s = 6.2 cm/s 5 1 km 5 km 0.75x10 9 m 3.76x10 9 m 0.075 m/s = 7.5 cm/s Table 1. Table 1 shows the critical shear when H 2 differs. One can see that the stratification structure (R d1 2 or R d2 2 ) is related to H 1 and H 2 in each layer, so by changing the thickness, we are changing the stratification as well. For deeper lower layers, the critical shear also increases. 3. Results and discussion 11
Fig 1. Eddy Energy for H 1 =1km, H 2 =1km Fig 2. Eddy Energy for H 1 =1km, H 2 =2km The background shear + 6 cm/s is much larger than the critical shear required for H 2 =1km and H 2 =2km, so we can see eddies develop fast within hundreds of days in Fig 1 and Fig 2. However, due to fast growth, it encounters numerical instability after 126 days for these two cases. Since we are more interested in the effect of varying layer thickness, we are not going to further investigate H 2 =1km and H 2 =2km. Fig 3a. Eddy Energy time series: Upper: H 2 =3km; middle: H 2 =4km; lower: H 2 =5km. Fig 3b. Upper/ lower eddy kinetic energy ratio: Red: H 2 =3km; blue: H 2 =4km; black: H 2 =5km. Fig 3a shows the eddy evolution of H 2 =3km, H 2 =4km and H 2 =5km, where the x-axis is days of simulation, y-axis is the eddy kinetic energy (Units: cm 2 /s 2 ) in the upper and lower layers. Eddy kinetic energy in the lower layer is always lower than the upper with no apparent phase shift observed. 12
According to Table 1, one may expect H 2 =3km to become unstable because the default background shear 6 cm/s satisfies the critical shear of 5.0 cm/s. Therefore, the abrupt growth of eddy kinetic energy seen in H 2 =3km within 300 days is expected. After this period of fast growth, H 2 =3km becomes stable with energy staying at 2 (Units: cm 2 /s 2 ). H 2 =4km and H 2 =5km are both below the critical shear, so we consider them as stable cases. However, we can still see energy growth even though it grows much slower (takes ~ 800 days to reach its peak value). Compared to the other two, energy growth in H 2 =5km is relatively slow and may still be developing at the end of the model run. Meanwhile, the energy level in H 2 =5km is one order smaller than H 2 =4km, and two orders smaller than H 2 =3km. So one can conclude that, as H 2 increases, the eddy kinetic energy in both layers becomes weaker (H 2 =4km ~ 1.5 cm 2 /s 2, H 2 =5km is ~ 0.15 cm 2 /s 2 ). Fig 3b reveals an interesting phenomenon concerning the energy ratio between two layers. We can see the upper and lower energy levels ratio increase as the lower layers deepen. Since H 1 is constant, the increase of ratio with increasing H 2 is probably due to less energy in H 2, rather than changes in the amount of energy in H 1. The following barotropic mode and baroclinic modes were calculated: Barotropic mode: ; upper/lower baroclinic modes: Fig 4. streamfunction of H 2 =3km (upper), H 2 =4km (middle) and H 2 =5km (lower). (Units: cm 2 /s) Left: Barotropic components; Right: Baroclinic components H 2 =3km and H 2 =4km show apparent zonal bands in both barotropic and baroclinic components in the last time step, but the jets in the barotropic part are stronger. H 2 =5km has not formed any bands during this period of time, and it looks like it is in the early noodle stage. The baroclinic part mostly resembles the barotropic structure for each experiment, however, the barotropic part is stronger for H 2 =3km and H 2 =4km. In H 2 =5km, we can barely tell which component dominates since they each have about at the same magnitude. The second baroclinic mode is not shown here since it is perfectly out-of-phase with the first baroclinic mode and five times weaker than the first one. 13
Fig 5. Zonal mean zonal wind for barotropic (blue) and two baroclinic (upper:red; lower: green) modes. Left: H 2 =3km; middle: H 2 =4km; H 2 =5km. (Units: cm/s) Fig 5 indicates that more zonal jets were observed in H 2 =3km than the other two. (Four waves for H 2 =3km, four waves for H 2 =4km and three waves for H 2 =3km) The perfect out-of-phase relationship between upper baroclinic mode and lower baroclinic mode can be seen in the zonal mean zonal wind as well. Unlike H 2 =3km and H 2 =4km, the barotropic mode and the upper baroclinic mode is about the same magnitude for H 2 =5km, but the former is usually stronger than the later for the other two cases. Fig 6. Mean zonal wind (upper/lower) and its shear of H 2 =3km. (units: cm/s) In the mean perspective, we can see four major pairs of zonal jets in H 2 =3km at both upper and lower layers. The zonal mean zonal shear (~ 2-2.5 cm/s) is strong as well. Fig 7. Mean zonal wind (upper/lower) and its shear of H 2 =4km. (units: cm/s) H 2 =4km shows one major eastward jet located at the center of the domain, surrounded by relatively slower westward flow. Also the zonal mean shear (~ 1 cm/s) is relatively weaker than H 2 =3km. 14
Fig 8. Mean zonal wind (upper/lower) and its shear of H 2 =5km. (units: cm/s) Similar to H 2 =4km, there is one major eastward jet for H 2 =5km. However, its intensity is much weaker and so is its shear (~ 0.2 cm/s). Compared to H 2 =3km, there is more zonal variability, which may be related to being under-developed. 4. Conclusion Table 1 shows the theoretical value of critical shear for the five experiments in unequal thicknesses channel model. For larger H 2, the stronger vertical shear is needed to achieve instability. However, critical shear is only the necessary condition for instability to take place, so we still need to make sure if the wave number lies within the long-wave and short-wave cutoff range. Other than instability, we examined the effects of different stratification structures on eddy energy evolution as well. By increasing the thickness of H 2, the eddy energy declined. It also takes longer for the energy to grow when H 2 deepens. The eddy kinetic energy ratio between upper layer and lower layer reveals that with deeper H 2, the ratio becomes larger. This is intuitive since potential vorticity becomes weaker when the thickness (H 2 ) is larger. In mean perspective, as H 2 increases, we see less zonal jets in the channel. The vertical wind shear also decreases with larger H 2. H 2 =5km has not reached its stable state since its kinetic energy keeps growing in the simulation, so we might need to extend the simulation period to make stronger conclusions. Part D: Mean Background Flow and Eddy Development in a Quasi-Geostrophic Equal Thickness Two Layer Model 1.1 Introduction The effect of mean zonal background (MZB) flow on eddy development is investigated in the framework of the theoretical Phillip s model where layers are adjusted to be of equal thickness. The MZB flow is varied and eddy activity in each layer is estimated based on eddy kinetic energy. It is observed that the layer in which this activity is stronger depends primarily on the direction of the MZB current and that for small values of the said flow eddy activity oscillates between layers 1 and 2. The theory of conservation of potential vorticity is invoked to explain the former while the latter is explained based on the solutions to the Phillip's problem. 1.2 Methodology The non-dimensional QG equations were solved in a channel model parameterized for oceanic flows in which the height of each layer was adjusted to be of equal thickness (H1=H2=H=2 nondimensional units). However, to maintain static stability is was necessary to adjust the Rossby internal deformation radius. This was accomplished using the relation: 15
From the above it is clear that R o 2 < H, as long as density and the coriolis parameter (modeled on a beta plane) are untouched. This resulted in an approximate value of 33 (non-dimensional units) for R o. In an attempt to model a purely baroclinic environment, a free-slip lateral boundary condition was imposed on the meridional boundaries of the channel, while boundary conditions in the zonal direction was periodic. The resulting QG potential vorticity equations are, for layers 1 and 2 respectively. Where 2 Ψ n is the relative vorticity and β o the gradient of planetary vorticity. A total of 13 runs were performed with U 0, the MZB current, varying from -6 cms -1 to 6 cms -1 (incrementing by 1 cms -1 ) and each run extending for 700 days. The MZB current is applied only to the top layer while the bottom layer is considered motionless. Hence, the absolute vertical shear of the MZB flow is equivalently U 0. The initial condition applied was a zonal wave number 15 disturbance with amplitude of 10 on the top layer and 2.5 on the bottom (model s default) and no meridional variation (figure 1). Figure 4: Initial Disturbance. A purely zonal disturbance with no meridional structure at day 1. 16
To calculate eddy kinetic energy, first the u and v components of the flow are calculated by taking meridional and zonal derivative of the stream function respectively. u=- Ψ/ y and v= Ψ/ x. These are further decomposed into zonal average and eddy components; u = [u]+u* and v = [v]+v*. Finally eddy K.E. = ½(u* 2 +v* 2 ). 2.1 DESCRIPTION OF RESULTS In figures 2, 3 and 4, time series of eddy kinetic energy covering the run period (top panels) is presented. These are calculated by taking a daily areal average of the eddy K.E. discussed above. It can be seen that runs with higher mean vertical shear (higher MZB flow) develop higher eddy kinetic energy. Moreover, for runs with -4 cms -1 < U 0 < 4 cms -1, the energy of the eddies in each layer go through several oscillations and eventually dampen out. The two bottom panels illustrate the distribution of the horizontal stream function corresponding to the eddy K.E. maximum for each layer. For runs with MZB current ranging from -1 cms -1 to 4 cms -1, the maximum K.E. occurs at the initial time, so the horizontal stream function plots for these runs correspond to the secondary maximum in eddy K.E. Figure 5: Regime 1 (U 0 =1 and 2 cm/s) the case U 0 =0 treated separately. Eddy K.E. in top and bottom layer almost perfectly out of phase. These figures confirm that the amplitude of the stream function is lower for cases of less background shear and highest for cases of higher background shear. Furthermore, it can be seen that the predominant zonal wave number of the initial condition prevails. But in addition to zonal waves one can now see meridional waves developing in the channel. These usually start from the 17
boundaries and propagate towards the center and the predominant meridional wave number is five. In the analysis to follow, special emphasis is placed on case U 0 = 0 (figures 5a-c) in an attempt to explain the apparent almost perfect 180 degrees phase difference between eddy activity in the two layers. This can be contrasted with the cases U 0 equal 5 and 6 cm/s where the response in both layers is almost perfectly in phase. In fact, one can designate three regimes. In regime one, U 0 < 2 cms -1, eddy activity between layers is approximately 180 degrees out of phase; regime two, 2 cms -1 < U 0 < 4 cms -1, is a mixed phase regime and regime three, U 0 5, 6 cms -1, activity is almost perfectly in phase. Additionally, one can see that in regime 3, depending on the direction of the MZB flow, activity is either stronger in the top layer or bottom layer. For eastward currents eddy activity is more predominant in the top layer, whereas for a westward flow the bottom layer is more dominant. Furthermore, eddies developing in the bottom layer from a westward flow are stronger than those developing in the top layer from an eastward MZB current. Figure 6: Regime 2. This is a regime of mixed phase 18
Figure 7: Regime 3, eddy K.E. at top and bottom layers are almost in perfect phase. Note also the shift in higher K.E. from layer 2 to layer 1 with westward to eastward mean zonal flow. 2.2 Analysis of Results It is well established that eddies develop to transport (density/temperature) anomalies meridionally so as to maintain the interface between layers in thermal wind balance. However, as eddies cause the flow to curve northward or southward, they are constrained by the conservation of potential vorticity (equations 1.21 and 1.22). Eddies are directly correlated with the relative vorticity. So it is fair to say that an increase in eddy activity corresponds to an increase in relative vorticity. It is then clear from equations 1.21 and 1.22 that for a westward directed MZB flow (beta term is appear in both so for difference it can be ignored in this analysis), the balance between the relative vorticity term and the isopycnal stretching term will mean that relative vorticity (eddy activity) can increase more in the bottom layers since in this case the last term is positive for the bottom layer. Conversely, for an eastward MZB current the opposite is true and relative vorticity can increase more in the top layer. 19
Figure 8: Case U 0 =0. Stream function and eddy K.E. perfectly out of phase. part b. plot of Eddy K.E. varying in layers while available potential energy (black) decreases monotonically. Part c, case U=0 plus no initial shear. K.E. in top and bottom layer equal and decrease monotonically. For the special case where U 0 =U t =U 1 -U 2 =0 (figure 5), the thermal wind is identically zero which characterizes a barotropic mean flow (Holton, 2004). To help elucidate the reason for the apparent 180 degrees phase shift we consider the solution to the Phillip s problem for which we can assume wave solutions of the form: Since our initial disturbance has meridional wave number equal zero we let l=0. Then the solutions to the coefficients A 1 and A 2 can be found from the simultaneous algebraic equations: 20
For non trivial solutions of the above system, the determinant must equal zero leading to a quadratic equation for the phase speed (not shown) into which U 1 =U 2 =0 can be substituted to yield: c=-β o /(k 2 +2/ R o 2 ) (this is one of two possible solutions) If this value for c is substituted into equations 2.21 and 2.22 and noting again that U 1 =U 2 =0, we find that A 1 = - A 2 for non-trivial solutions, which translates to a 180 phase shift in Ψ 1 and Ψ 2. Physically this corresponds to the system being released from rest where potential energy is maximum (determined from initial condition). Since there is no background flow from which eddies can gain energy, they simply consume energy from the initial tilt (shear) which rapidly decays. As shown is figure 5b, the available potential energy decreases monotonically in this process. Figure 5c, confirms that the available potential energy is directly related to the shear. Moreover, the eddy K.E. In layer 1 and 2 decrease monotonically, implying the oscillatory growths observed in the case of an initial tilt are indeed due to eddies feeding on that initial tilt/shear. 3.1 Conclusion The experiments confirmed that eddy development in the two-layer QG system is directly related to the vertical shear in the mean zonal background (MZB) flow. It was also shown that depending on the direction of this flow, eddy activity is higher in the top layer or the bottom. Eddies grow by extracting energy from the shear of the mean background flow. When this shear is positive corresponding to an eastward flow in the top layer, conservation of potential vorticity dictates that relative vorticity (a metric for eddy kinetic energy) can increase more in the top layer. Conversely, when the shear is negative (westward top layer flow) the opposite holds. For very small values of the background shear, the initial disturbance decayed rapidly with eddy activity in the layers almost perfectly 180 degrees out of phase. For a zero MZB which corresponds to a barotropic mean state a high in the top layer must balance a low in the bottom layer and vice versa explaining why eddy activity must oscillate between the two layers at 180 degree phase difference. In the case of zero MZB with initial vertical shear the resulting initial disturbance decays but this vertical shear provides a source of energy for eddies to develop in alternating layers while the disturbance dampens out. If no initial shear is present the disturbance dies monotonically in both layers presumably due to viscosity. Part E: Two Layer System Baroclinic instability theories are among the most popular subjects in geophysical fluids dynamics. Some of those theories focus on the study of continuously stratified fluids, like the Eady and Charney problems. For this project we will use a simple two layer model, which is more suitable for the ocean, to demonstrate the effect of baroclinicity. This problem was first studied by Phillips (1951). In the two layer system, the ocean or atmosphere are represented by two discrete layers, therefore only the barotropic and first baroclinic modes will be present in this system. To simplify the problem even more, a constant vertical shear will be used in each layer, this still allow us to analyze baroclinic instabilities. The mean flow is only in the zonal direction and constant with latitude. One remarkable difference between the two layer model and the Eady model is that a constant shear in the mean flow produces a mean potential vorticity gradient 21
along with the beta effect. This can be seen by equation (1), which is not the case for a continuous stratified fluid. where λ is the inverse of the Rossby radius of deformation. For a quasi-geostrophic two layer model, vertical shear in the mean flow will be related to horizontal density or temperature gradient by the thermal wind relationship. For the simpler case where the mean flow is only zonal, the isopycnal or isothermal surfaces will have a slope in the north-south direction. Given that the top and bottom boundaries are assumed to be rigid, from equation (1) we see that a stronger mean flow in the top layer will gives a positive mean PV gradient for the top layer and a negative mean PV gradient for the bottom layer. If the β effect is small enough, such change of sign in the PV gradient satisfy the Charney-Stern condition for instability. The problem: For this part of the study we will use a two layer quasi-geostrophic model to validate the importance of vertical shear and wavelength for the development and growth of baroclinic instabilities. Three different initial perturbations will be introduced in the model, all with the same amplitude of -2cm for the top layer and -0.5cm for the bottom one. The first (see Figure 1.a), will be long wave negative perturbation on the streamfunction in the model. This initial perturbation will have a wavelength longer than the long-wave cutoff limit, calculated by equation (2). This perturbation will have a supercritical shear of 11cm/s, with a mean velocity of 6cm/s on the top and -5cm/s on the bottom layer. The critical shear will be calculated from equation (3). This perturbation should be on the stable side of the stability diagram (Figure 2), hence it should not grow even for a strong enough shear. The second case (Figure 1.b) will be a short wavelength disturbance smaller than the short wave cutoff limit. For this case a shear of just 1 cm/s, or a mean flow of 1cm/s for the top layer and no flow for the bottom layer will be used. As the first perturbation, this will also be on the stable side of the profile, (Figure 2), but in this case, both shear and wavelength will work to prevent the grow of this disturbance. We expect that in this case, the perturbation should last only for a short period of time. For the third case, (Figure 1.c), a perturbation with wavelength between the short and long wave cutoff limits will be introduced under a shear of 6cm/s, or 6cm/s for the top layer and no flow for the bottom one. Here, both the shear and the wavelength should help the system to develop, it will be on the unstable part of the stability curve, (Figure 2), therefore we expect some growth until a new equilibrium is reached. Where R is the Rossby radius of deformation, and K is the wavenumber. 22
Figure 1: Initial perturbation (top layer) of the model for a) long wavelength, b) short wavelength, and c) critical wavelength. Note that the entire domain is represented on this figure, therefore the small perturbation will be very difficult to see. The critical shear of the system is determined by the following expression: From equations (2) and (3), we see that the planetary vorticity gradient has a strong dependence on the wavelength. The stabilizing effect of β increases as the wavelength increases. Static stability or stratification also have stabilizing effect but on the short wave limit. 23
Figure 2: Neutral curve for the two layer system showing the 3 cases studied in this project. The x-axis is the nondimensional wavenumber, and the y-axis is the nondimensional vertical shear. From figure 2, the only case where we expect some growth is in the third case. To demonstrate this, we ran the model for a period of 500 days after the initial perturbations were introduced. In figure 3, we show the time evolution of each cases for both the top and the bottom layers during a 500 days run of the model. The bottom 2 panels are just a continuation of the previous two panels for the unstable case 3. Therefore, we ran the model for 1000 days only for the unstable case 3, since this is the only that shows growth with time. All cases where initialized with the same amplitude of -2cm and -0.5cm perturbations for the top and bottom layers respectively. From figure 3, we can obtain valuable information. The first thing to note is that for case 2, or the short wave, the perturbation decays to nothing in a few days, this was expected, since both the shear and wavelength are not favorable for growth. For case 1, the long wave perturbation also diminished with time. The positive signals developed are just a response of the system to the initial perturbation in an effort to restore balance. From the last four panels of figure 3, we see a strong growth in time for the amplitude. 24
Figure 3: Time evolution averaged for the entire meridional width of the channel, for the top and bottom layers for each case. The bottom 2 panels represent a continuation of the previous two, i.e. the 500 to 1000 days run of the model for the unstable case. U represents the vertical shear for the run. This figure shows from day 5 on. Note that the contour intervals are different for the first 500 days to the last 500 days of the unstable case. Also is good to note that the growth reach a maximum at about 850 days of the initial perturbation. After that point the system stabilizes. To better represent this, the area average of the amplitudes for each cases are plotted on figure 4, each plot is a function of time. Again, the bottom two panels are just a continuation of the previous two. 25
A steady decrease in amplitude throughout the domain is showed in the long wave or case 1. The reason for the amplitude values to be so low is because an area average of the entire domain was taken. We performed this averaging to better demonstrate that under such conditions no growth, in the case 1 or 2, or growth in case 3 would occurs. For the top layer in the short wave case, a steady decrease in amplitude is not as remarkable as in the other cases because that same area averaging may be contributing to introduce small amplitude noises in the plot. But it is clear that case 3 or the critical wavelength is the only experiencing steady growth from the initial -2cm and -0.5cm perturbations introduced in the top and bottom layers respectively. From the bottom two panels of figure 4, we see that the growth stops at around day 800 from the initial run. This equilibrium state may be referred to as geostrophic adjustment, since this model is based on QG theory, geostrophy is one of the big assumptions. A good thing to note from figure 3 is that the speed of propagation of the perturbation is westward in the absence of a mean flow, this is consistent with the Rossby wave phase speed. Also, the propagation speed increases westward as the wavelength of the perturbation increases, again consistent with equation (4). for barotropic case, and (4) for the baroclinic case. In this case, U is the background flow, k is the wave number, this is different for all the three cases, lambda is the inverse of the Rossby radius of deformation, which for all the cases studied here a value of 25 km was used, beta = 2E-8 /(km s), and C is the phase speed. On figure 5, we show the time evolution of the perturbation for case 3 in both layers. The first panel, (day 100), show that more waves are being produced by the initial wave. The amplitude is actually decreased as compare to the initial perturbation. This may be due because the waves are propagating energy to the rest of the domain faster than their growth rate. By day 400, the meridional noodles have developed, but they have an arc shape because no slip conditions were applied on the north and south boundaries. By day 700, on figure 6, the noodles are developing into eddies. This is about the point when the eddies and mean flow are interacting more energetically. Further discussion on this will be presented ahead on this paper using the Eliassen- Palm Flux theory of wave mean flow interaction. By day 800, the perturbations have reached their maximum amplitude, and an equilibrium state has been set. After this time, just oscillation about the equilibrium is present. 26
Figure 4: Area average of the amplitudes for each cases as a function of time. Note that for the long wave (case1), and the short wave (case2), the amplitudes experiences a steady decrease to zero. Only the critical wave (case 3) shows steady grow with time. The grow stops as the time increases, subjecting a balance was reached. The x-axis has a time increment of 50 days. 27
` Figure 5: Snap shots for the top (left panels), and bottom (right panels) layers for the times given in the figure s title. Each time are based on the number of days after the initial perturbation was introduced. 28
Figure 6: This plot follows from figure 5. Results: The simple two layer model was demonstrated in this study by introducing three distinct perturbations with different wavelength under different shear. The first case was a long wave, larger than the longwave cutoff limit, under a shear strong enough to support instability. According to the stability theory, this perturbation should not grow, as it was demonstrated. For the second case, a short wave, shorter than the short-wave cutoff limit was put under a shear that was subcritical. Again this perturbation was under strongly unfavorable conditions for growth. As figures 3 and 4 showed, the perturbation did not last for a very long time. In the third case, a wavenumber anomaly in between the short and long wave cutoff limit under a favorable shear was introduced in the model. As expected this perturbation grew exponentially to hundreds of times larger than the initial one, but a balance was reached after about 800 days from the initial perturbation. All this totally agrees with the Phillip s model and theory for baroclinic instability. 29
Part f: Wave Mean Flow Interaction In this section it will be analyzed the simulation made in the previous experiment; the energy supplied from the background flow to develop Rossby waves from an initial perturbation is followed In this process the mean flow suffers a deceleration and the perturbation an exponential growth. This analysis immediately leads us to the Eliassen-Palm flux vector (E). In figure 1 it is shown the evolution of the total anomaly in the pressure field. All the figures in this section refer to the bottom layer, as the ones of the top layer are very similar. The period of simulation was one thousand days. During more than half of the period the perturbation is imperceptible, then the vortex got unstable around the six hundredth day. At that point, its evolution appears to be a cascade down to an increasingly disturbed state of the whole channel. [X: Time in days, Y: Value] Figure 1. Evolution of the pressure field anomaly in the first layer. The E vector is a two-dimensional field in the latitude-pressure coordinate (equation 2, standard notation). The first component of E is the flux of momentum in the northward direction; this process is a barotropic mechanism and is analogous to the Reynolds stress. The second component measures the flux of buoyancy in the vertical direction; this is the baroclinic part of E. Due to this property this component is also referred to as the form stress component....(1) Equation 2 states that the pole-ward eddy flux of potential vorticity is related to the E vector via the horizontal velocity and temperature/density anomalies. In effect, the modification of the 30
mean state by eddies is described by the divergence of E. However this result is quite general; it can be equally valid for steady or transient disturbances provided the quasi-geostrophy be valid (James Ian, 1995). In this simulation there are no steady eddies so it is for sure that only transient eddies are the ones acting over the mean current....(2) From equation 3 the tendency of the mean flow is balanced by the divergence of E, friction and the residual Coriolis force denoted by v*. The divergence of E represents a force per unit mass on the mean flow which acts on the tendency of the PV in the current, so the joint effect of the right hand terms of equation 3 are associated to the acceleration/deceleration of the mean flow in the channel....(3) At this point we could guess that the most interesting things happened before reaching the e folding point in figure 1. In our experiment the perturbation began to grow nearly after reaching 600 days of simulation (figure 1). At this time the flux of momentum and buoyancy reached the maximum (figure 2). Once the maximum flux is reached, near in the 750 day, the flux of momentum and buoyancy begins to decay quickly. The pump on the initial incipient perturbation takes place in relatively short and abrupt period rather than a prolonged and smooth way; this is a rather expected result from the non-linear dynamics that involve this process. [Reynolds Stress Component of E] a) [X: Time in Days, Y: Latitude] b) [X: Time, Y: Height] [Form Stress Component of E] 31
c) [X: Time in Days, Y: Latitude] d) [X: Time, Y: Height] Figure 2. Zonally averaged evolution of the E components. a) momentum flux component in the lower layer and b) a cross section. c) buoyancy flux component in the lower layer and d) a cross section. A very interesting characteristic of the E vector is that its contours are parallel to the contours of the group velocity wave field. Also, a quantity called wave activity (A) can be defined thru the divergence of E (equation 4), which in absence of dissipation forces is a conserved quantity...(4) The E vector field (figure 3) provides information about the energy propagation of the Rossby waves, the divergence gives information about the generation, dissipation and mean flow interaction of the disturbances (James, I, 2). This is a remarkable property which has made the E vector applicable in several different topics in meteorology. So, the westward propagation of the energy depicted by the E vector in figure 3 and 4 corresponds to the propagation of the group velocity. After the Rossby waves were formed, this energy propagation can be associated with short waves, as its direction is eastward. 32
[X: Longitude, Y: Latitude] a) b) [X: Longitude, Y: Latitude] c) d) Figure 3. Snap shot of the magnitude field of E at a) 200 days, b) 400 days, c) 800 days and b) 1000. From equation 3, negative divergence of E suggests a deceleration in the mean flow, that is, convergence of mean flow energy into the vortex. That means that the perturbations are the medium of the flow for taking out its unstable energy, then reducing its sharp gradient to a final equilibrium state. The maximum convergence in this experiment takes place between the 600 and 800 days of simulation (figures 4 and 5), days in which the Rossby waves were formed. 33
[X: Time days, Y: Value] Figure 4: Zonal and Meridional averaged components of the divergence of E; black line: form stress component and green line: Reynolds stress component. [X:Time in days, Y: Latitude] Figure 5. Evolution of the divergence of E vector. Vectors pointing left are negative values. 34
a) b) c) [X: Latitude, Y: Height] Figure 6. Cross-section of the divergence of E at a) 600 days, b) 700 days and c) 800 days. Vectors pointing down are negative values. Figure 6 shows the zonal and meridional averaged time series of the buoyancy and momentum components of the divergence of E. The buoyancy flux convergence starts first with a rapid increase suggesting that the growth process is at the beginning stimulated from the baroclinic mechanism. The momentum flux exhibits a changing sign behavior before and rapidly after its maximum. Convergence to and divergence from the perturbation suggests that the Reynolds stress plays a more important role in the beginning of stabilization of the mean state of the current. Nearly after the 800 day the flow, after the maximum transfer of energy, has almost stopped (figure 2), the divergence of E begin to weaken, and the Rossby waves begin to dissipate in the mean current transferring some of its eddy kinetic energy back to the mean flow towards an equilibrium state. Conclusion: Several experiments were conducted using a two layer quasi-geostrophic channel model parameterized for oceanic flows. The first set of experiments focused primarily on the effects of varying specific parameters of the model while in the last couple of experiments, perturbations were introduced into the model's flow. In the first experiment, which concentrated on investigating the importance of eddies in momentum and heat transport, it was found that transient eddies are the primary mechanisms through which such transports occur and that this transport was directed in such a way as to transport warm water poleward and cold water southward thereby reducing the isopycnal gradients through conversion of available potential energy to kinetic energy. It became apparent that for an eastward flow, the top layer developed stronger eddy activity and that for a westward flow the opposite holds. 35