Dispersion of Energy from the Mixed Layer due to Near-Inertial Waves

Transcription

1 Dispersion of Energy from the Mixed Layer due to Near-Inertial Waves Patrick Meyers MPO 611 May 8, Introduction It has been well established that atmospheric forcing creates inertial currents in the forced region (Pollard, 1970). When these currents are excited, they create a set of near-inertial gravity waves which propagate away from the storm region. In the days and weeks following the storm, the ocean s baroclinic response causes energy propagation down and out of mixed layer. Some of the energy in transferred into the pycnocline, while some propagates equatorward and out of the forced domain. Dispersion alone is not strong enough to account for the observed decay of the inertial oscillations, so it is expected that higher frequency oscillations and destructive forcing by the atmosphere play a role in dampening the oscillations (Pollard, 1980). This paper will examine the different parameters that influence the energetics of the near-inertial waves. The second section examines inertial oscillations and the formation of these baroclinic waves. Section 3 considers the landmark paper on baroclinic wave response to forcing events by Gill (referred to as G84). Section 4 reviews other modeling studies of energy propagation out of the mixed layer. 2. Inertial Oscillations As a strong wind event travels over the ocean, a transfer of momentum occurs between the air and atmosphere, inducing an ocean surface current. These currents become well established as quickly as five hours after the onset of wind forcing (Chang and Anthes, 1978). Once the atmospheric forcing stops or the storm has moved out of the region, the currents maintain their original momentum and continue to flow inertially. In the presence of the earth's rotation, these currents follow a clockwise (counterclockwise) path in the northern (southern) hemisphere with a frequency very close to the Coriolis parameter f=2ωsin(θ). These trajectories are advected with the background flow.

2 The simplified equations that govern these motions in the absence of non-linear dynamics and dissipation in the mixed layer are: u 1 p fv = t ρ x v 1 p + fu = t ρ y (1) (2) For pure inertial flow, the RHS of (1) and (2) are set to zero. However, for realistic oceanic flows the horizontal pressure gradient induced by these currents cannot be ignored. It is these slight modifications by the pressure gradient force that causes these currents to be near-inertial. In general, the observed frequencies are around 1.01f-1.10f (Price et al, 1981). The solution to these equations appears as sinusoidal function, with the phase of meridional velocity about 90 ahead of the zonal velocity. Figure 1 shows drifter trajectories following the passage of Hurricane Gustav in the Gulf of Mexico from the 2008 hurricane season. It is easy to identify the loop-like inertial oscillations in the forced region. The radius of the loop is directly proportional to the strength of the surface current. Inertial currents are most likely to form in regions where the relative vorticity of the winds is the same sign as the relative vorticity of the inertial currents (Price, 1981). In the northern hemisphere, this implies that regions of negative relative vorticity induce inertial oscillation. This explains why stronger inertial oscillations are observed on the right side of northward propagating tropical cyclones. Behind a passing wind forcing, a wake of inertial currents and internal waves is formed in the mixed layer. The inertial currents that are formed at any one point will be slightly out of phase with currents around it. This creates an interference pattern in the wake of the storm consisting of Figure 1-Drifter trajectories following Hurricane Gustav from 2009 showing inertial oscillations. (Courtesy of Rick Lumpkin) 2

3 oscillating converging and diverging inertial currents. These zones of convergence and divergence cause a vertical displacement of isopycnals, which implies horizontal pressure gradients, and therefore a current in the thermocline. The frequency of the up and down displacement of the thermocline occurs at the inertial period, f. A schematic drawing of this inertial pumping is shown in figure 2. The signature of these oscillations can be seen in time series of temperature profiles in the wake of a forcing event. Figure 3 shows the time evolution of the upper ocean temperature profile following the passage of Hurricane Gustav. Since salinity gradients are very small in the Gulf of Mexico, then isopycnal surfaces can be approximated by isotherms. These observed fluctuations in temperature are caused by inertial pumping. In the presence of such stratification, inertially induced internal gravity waves will propagate away from areas of deformation of the pycnocline (G84). The adjustment problem will be addressed in the next section, discussing the assumptions and theory of G84. Figure 2 The inital stages of adjustment. A storm is presumed to move rapidly in the x-direction leaving behind (a) a current in the mixed layer. At this stage the thermocline and deep ocean are unaffected and remain at rest. (b) The adjustment process starts with inertial rotation of the current in the mixed layer. This causes convergence and divergence, and therefore inertial pumping, at the base of the mixed layer (G84). 3. Gill's assessment of internal waves in the wakes of storms The motivation for Gill's modeling experiment was mostly due to the inadequacy of previous modeling studies on the baroclinic adjustment following storms. Rossby's original paper on readjustment (1938) did not consider transient flows, even though at large scales they carry most of the energy. A follow-up study by Bolin (1953) attempted take into consideration these transients, but the potential vorticity injection was distributed over the entire depth of the fluid. This was corrected by Pollard (1970), when 3

4 Figure 3- Temperature profiles from thermistor chain data following the passage of Hurricane Gustav at time 0. Inertial pumping can be seen as the oscillations of mixed layer depth (Courtesy of Rick Lumpkin). he restricted the input of PV to the mixed layer. From observations, Pollard saw that there was a significant energy exchange between the mixed layer and thermocline, which he parameterized by an empirical decay coefficient. Gill's paper aims to better understand the energy exchange out of the mixed layer in order to model them more accurately. A. Equations and parameters Before discussing the results of Gill, it is first necessary to understand some of the assumptions and considerations built into the model. The model consists of an ocean of uniform depth with a Boussinesq approximation. The hydrostatic approximation has also been made. Gill takes into considers a storm moving zonally, such that the induced current is independent of x, therefore restricting the varying response to the y direction. The full equations of motion, including equations (1) and (2) simplify to: (3) (4) (5) (6) Where the equations depend on a particle's meridional displacement η, the vertical displacement h, Brunt-Vaisalla frequency N, and the step function S(z) which restricts the currents to the depth of the mixed layer. One important note to make about the Gill model 4

5 is that it ignores the barotropic response to the storm because the wave response occurs nearly instantaneously and the associated currents are very weak. This study can be made more transparent by expanding the parameters into vertical normal modes. The form of the expanded variables is: (7) (8) where the tilde denotes variables that are a function of y and t, and the hat represents normal-mode eigenfunctions when n 1. (9) (10) where the eigenvalue c n represents the wave speed of the nth mode. Substituting the normal modes into the equations of motion leads to a differential equation for η which is determined by the initial current: (11) Some important properties of the solution can be obtained by examining a periodic, large scale case on the f-plane. The initial current has the form u=sin(ly). This results in solutions that are dependent on y, z, and t. (12) where σ n represents the vertical profile step function and ω is the square of the frequency of the nth mode with the dispersion relationship (13) This solution clarifies some relationships between the initial parameters of the problem. The properties of each mode rely heavily on the dispersion relationship, namely the ratio between the horizontal scale and the Rossby radius. f : lc n...(14) 5

6 For synoptic scale conditions (l -1 =1000km, c n /f=30km), this ratio is 33. Therefore it can be seen from (13) that the internal mode has a frequency that is just slightly greater than f. As you move to higher order modes, the wave speed decreases, and therefore the ratio increases, meaning that the frequency for higher order modes becomes even closer to the purely inertial frequency f. B. Vertically propagating energetics It is possible to use the normal mode expansion discussed above to get a better understanding of the energy lost from the mixed layer. First, consider the time that it takes for the nth baroclinic mode to become 90 out of phase with the pure inertial oscillations. (15) As the time moves closer to t n, energy is lost from the mixed layer and the current in the mixed layer weakens. The rate of this energy lost is clearly related to (14), such that faster energy loss occurs at smaller values of the ratio so there is a significant increase of the near-inertial frequency. At typical synoptic scales, the time to be 90 out of phase is on the order of one year for the first baroclinic mode. Therefore it is unrealistic to expect much energy propagation from the mixed layer to the deeper ocean for normal atmospheric conditions. With tropical cyclones, energy loss can occur on time scales on the order of days because of relatively large Rossby radii at low latitudes. The current in the mixed layer can be defined by the time dependent function where A depends on the time and vertical evolution of the summation of the vertical modes. The energy of the mixed layer can be approximated by A 2. Figure 4 shows the amplitude of the inertial oscillations in the mixed layer for a case of parameters similar to that expected in a hurricane. After about (16) Figure 4 - The amplitude of the inertial oscillation is modified by the interference created by the difference baroclinic modes. The first four modes are shown here (G84). 6

7 13 days, there is a minimum in energy of the mixed layer, before the energy then rebounds, but never reaches its original amplitude. This suggests energy propagation out of the mixed layer to the pycnocline. The interference pattern between the different frequencies of normal modes creates an inertial beating, an oscillatory pattern of nearinertial current amplitude. Looking at the time evolution of the zonal velocity with depth reveals some characteristics of the energy fluxes. Figure 5 shows a clear propagation of zonal velocities out of the mixed layer in the first fifteen days after a forcing event. There is apparent upward phase propagation near the surface, which is consistent with a transfer of energy downwards. These induced currents in the top 1000m reach an amplitude of only about 5% of the mixed layer current. Most of the kinetic energy is still held in the mixed layer. There is a descending energy ramp in the first four days, and a similar one between 10 and 15 days, which is consistent with the amplitude seen in figure 4. Figure 5 Zonal velocity u as a function of depth for the 15 days following forcing. Contours range from to m/s with 0.05 m/s contours. Red arrows indicate the downward propagation of velocities associated with inertial beating. 7

8 C. Beta-plane effects Not only is energy dispersed vertically, but there is also horizontal dispersion of waves. The waves will move laterally at the group velocity (17) Because the waves that are generated are near-inertial and close to their turning latitude, latitudinal changes of the Coriolis parameter need to be considered. As a wave propagates northward, it will eventually reach its turning latitude, at which point it is deflected southward. Southward propagating waves are accelerated south by changes in the Coriolis parameter. Figure 6a shows the propagation of height anomalies from the storm. Three distinct bands can be seen, which are bounded by the first three baroclinic wave modes that initially traveled northward or southward. The ray paths of the first three modes can be seen in figure 6b. The time scale for the energy to propagate out of the atmospherically forced region is on the same order as vertically propagating waves out of the mixed layer. Figure 6 (a) Plot of h(y,t) 100m below the mixed layer for a storm centered around 2700km north on the beta-plane. (b) Ray paths for modes 1 (solid), 2 (dashed), and 3 (dotted). (G84) 8

9 D. Conclusions and Discussion Gill's analysis of the energetics of the mixed layer highlights the importance of vertically and horizontally propagating waves. For large scale, weakly forced events, the phase and group speed of the baroclinic waves is small, meaning that most of the energy is retained in the mixed layer. For stronger, smaller, faster moving storms, energy is dispersed from the mixed layer much faster, transferring energy and momentum into the upper thermocline. One key process that is ignored in Gill's modeling efforts is the dissipation associated with turbulent processes. It has been well established that storm-induced mixing occurs during strong forcing events (Chang and Anthes, 1979). Vertical shear of the currents causes mixing of cooler thermocline waters into the mixed layer, which acts to decrease the kinetic energy because of viscous dissipation and the increased potential energy from mixing. 4. Other modeling and observation efforts In this section, other studies of near-inertial waves will be examined in light of the previous work by Gill. Seasonal variations in energy flux to the thermocline were studied by D'Asaro (1985). The modeling efforts all consider isolated storms, rather than realistic 'random' forcing. Inertial currents can be enhanced or destroyed by subsequent wind forcing. D'Asaro studied the seasonal variability of inertial energy fluxes into the mixed layer by fronts, lows, and other forcing that occurred over oceanographic buoys. Energy flux through the mixed layer peaks in October, during a time of deepening of the mixed layer by surface cooling and an increased number of storms that occur in the winter season (Fig. 7). One of the significant finding from this paper is that the energy flux from the wind to inertial motions in the mixed layer are of similar magnitudes to the wind flux forcing of mixed Figure 7- Ten year monthly average of inertial energy flux through the mixed layer with bars layer turbulence. This mixing is another sink denoting the standard deviation (D Asaro). 9

10 of energy transferred from the atmosphere to the mixed layer. Kundu and Thomson (1985) found similar results as Gill, with energy decaying in the mixed layer over time as currents are induced at depth. In contrast to Gill, they considered a propagating storm front. The moving front creates a modification to the inertial beating because inertial currents are not created simultaneously. Broutman and Young (1986) demonstrated that high frequency internal waves can extract energy from near-inertial waves. These high frequency waves are enhanced as the near-inertial waves pass through, resulting in a decreased amplitude of the near-inertial waves. While this phenomenon is not readily observable from observations, it is expected that such an interaction would occur at the base of the mixed layer in the highly stratified pycnocline. Modeling of near-inertial waves was expanded into both horizontal directions by Levine and Zervakis (1995). The model results support many of the conclusions of G84. They found a transfer of energy from the mixed layer to the pycnocline around the time it takes the first baroclinic mode to become out of phase with the pure inertial oscillations (Fig. 8). There is also a total decrease of energy of the water column around the time it takes for the laterally propagating waves to exit the region. Due to the vertical structure Figure 8- The top panel displays total energy (thick line), mixed layer energy (thin line), and pycnocline (dotted line) on the beta-plane as a function of time.the middle panel shows the u component of the mixed layer current. The bottom panel shows the horizontal current vectors. Contours of current speed are superimposed on top (Zervakis and Levine, 1995). 10

11 of the baroclinic modes, once the wave departs the area the energy in the mixed layer is decreased, but not uniform over all depths. It was also found that the beta effect causes an acceleration of the inertial beating frequencies. 5. Summary This paper discussed the implications of near-inertial waves that are generated in the mixed layer after strong atmospheric forcing events. Storms induce currents in the mixed layer, which create oscillating areas of convergence and divergence over the days following the storm. The inertial pumping creates waves in the pycnocline which will propagate both vertically and laterally. It is possible to decompose the oceanic response into a series of baroclinic modes, which characterize the ocean structure and its evolution in time. The induce inertial-gravity waves have a frequency slightly faster than the pure inertial frequency, f. This creates an interference pattern over time, as the different modes become out of phase. This inertial beating is characteristic of energy transfers between the mixed layer and the pycnocline. As energy slowly propagates downward, weak currents are created at depth. There is also significant lateral dispersion of energy as the baroclinic waves depart the atmospherically forced region. The beta effect also plays a critical role in turning waves equatorward. This effect acts to increase the rate at which energy is removed from the mixed layer. The dispersion of energy is not completely accounted for by these near-inertial waves. Much of the dissipation of the surface currents is due to random forcing by surface winds (Pollard, 1970). Further dissipation of energy is due to shear-induced mixing that occurs at the base of the mixed layer (Chang and Anthes, 1977). 11

12 Works Cited Bolin, B., 1953: The adjustment of a non-balanced velocity field towards geostrophic equilibrium in a stratified flow. Tellus, 5, Broutman, D. and W. R. Young, 1986: On the interaction of small-scale oceanic internal waves with near-inertial waves. Journal of Fluid Mechanics Digital Archive, 166, Chang, S.W. and R.A. Anthes, 1978: Numerical simulations of the ocean s nonlinear, baroclinic response to translating hurricanes. J. Phys. Oceanog., 8, Chang, S.W. and R.A. Anthes, 1979: The mutual response of the tropical cyclone and the ocean. J. Phys. Oceanog., 9, D'Asaro, E.A., 1985: The energy flux from the wind to near-inertial motions in the Surface Mixed Layer. J. Phys. Oceanogr., 15, Gill, A.E., 1984: On the behavior of internal waves in the wakes of storms. J. Phys. Oceanog., 14, Kundu, P.K., and R.E. Thomson, 1985: Inertial oscillations due to a moving front. J. Phys. Oceanogr., 15, Pollard, R.T., 1970: On the generation by winds of inertial waves in the ocean. Deep-Sea Res., 17, , 1980: Properties of near-surface inertial oscillations. J. Phys. Oceanog., 10, Price, J.F., 1983: Internal wave wake of a moving storm. Part I: Scales, energy budgets and observations. J. Phys. Oceanog., 13, Rossby, C. G., 1938: On the initial adjustment of pressure and velocity distributions in certain simple current systems II. J. Mar. Res., 2, Zervakis, V. and M.D. Levine, 1995: Near-inertial energy propogation from the mixed layer: Theoretical Considerations. J. Phys. Oceanog., 25,