Potential and Kinetic Energy in the
equatorial
The equatorial Pacific can be thought of in terms of kinetic energy and potential energy. The kinetic energy is the movement energy in the ocean currents. The potential energy comes from the slope of the isopycnals. That is, unless the isopycnals are flat then their slope is storing up energy, this is referred to as available potential energy (Lorenz 1955).
1. Kinetic Energy
The equation for the time rate of change of
kinetic energy is derived from the three dimensional momentum equations
(discussed in
where K represents kinetic energy, is the 3-dimensional velocity and is the 2-d velocity (u,v). The represents deviation from the horizontally averaged field, in the case of pressure the field not in hydrostatic balance with the reference density field, ps is the surface air pressure.
The terms on the right hand side can be understood as (1) the KE through the walls of the system, (2) the work done against the pressure gradient by the ageostrophic flow, (3) the KE that is lost to vertical movement of the mass field (buoyancy power), (4) the windstress on the currents and (5) and (6) are the friction from horizontal and vertical shears.
The primary balance of the KE equation however is
This is the energy representation of the wind maintaining pressure gradients in the ocean. The work done against the large pressure gradient in the equatorial Pacific can be separated out from the buoyancy power. Goddard and Philander (see their Figure 5b) found that the pressure term over 15° S to 15° N is always a sink (which is why they chose this region). The only source of energy therefore is the wind. Note that they do not say that the wind term balances the buoyancy term, the pressure term is still significant but always negative. Perturbation energy gained by the ocean from the wind will be radiated out of the volume through the pressure power or dissipated by wave diffusion.
2. Potential Energy
The available potential energy is given by Oort et al. 1989 as
The rate of change of potential energy can be written as:
where P is the available potential energy (Oort et al. 1989) which refers to energy that may be stored or extracted by a redistribution of the mass field (Lorenz 1955). The terms on the right hand side can be thought of as (1) the advection of APE through the wall of the volume, (2) the vertical motion of the mass field, (3) the apparent source or sink due to shear in the stability profile, (4) the horizontal diffusion of APE through the walls of the volume, (5) the dissipation of APE within the volume due to horizontal density gradients, (6)vertical diffusion and dissipation of APE and (7) the change of APE due to surface fluxes of density connected with thermal and fresh water sources.
For the APE equation the primary balance is simply
Goddard and Philander demonstrate the accuracy of this equation in their Figure 5a. Largest discrepancies occur at the peaks. There are no sources or sinks in this equation, it merely describes a redistribution of mass. Heat fluxes were shown to be small. At the end of some large events,there is a sizable amount of energy flowing through the eastern boundary of the domain.
3. Combining potential and kinetic energy
The potential and kinetic energy have a common term, representing the conversion from KE to APE. Equations and can be combined,
relating the potential energy to the wind work done on the ocean.
The time derivative of APE and the wind work are shown in Figure 1. The wind work and time derivative of APE correlate well but the wind is supplying more energy than the APE is receiving. Energy is being lost somewhere, most likely to the pressure term as shown by Goddard and Philander.
How do I calculate this pressure term,
what does it really mean??? It seems that it would account for most of alpha.
The meridional winds were also included in the calculation (red line) and have a minimal contribution to the balance.
Figure 1 The
energy balance for the GFDL model according to equation . Blue line is the time
derivative of APE, red line is the zonal wind work, green line is the total
wind work including meridional winds.
In Fedorov (2006) the work energy was calculated from the surface zonal windstress and zonal currents in the top 30m according to
The perturbation wind energy is made up of four terms:
(1) (2) (3) (4)
The overbar here represents climatology and the prime the interannual variability.
The zonal perturbation work term is created from the sum of terms 2, 3 and 4 in equation , (Figure 2b,c,d respectively). Each of these terms contributes. Term four has large peaks related to El Niño events (green line shows Nino3.4) and is small otherwise. The large peaks occur when the interannual anomaly of windstress and surface zonal velocity are high in an El Nino due to decreases trades and increased SEC. Note that this term is always positive which means that the windstress and zonal velocity change sign at the same time (as would be expected). It is interesting that a corresponding peak doesn’t appear for La Niña events. Term 4 has been smoothed with a 12 month running mean; when this isn’t done, the peaks are about twice as large.
Terms 2 and 3 are combinations of climatological and interannual anomalies. Term 3 is relatively small. Note that peaks in term 4 are counteracted by peaks of the opposite sign in 2 (Investigate this further to figure out why this is so).
What exactly is the
work term measuring? The amount of wind energy that is being
felt by the surface layer of the ocean? If so, away from the equator
would zonal windstress correspond to meridional flow?
In the previous section we showed that the meridional wind term is negligible. Here we show it term by term (red lines in Figure 2).
The peaks in the meridional term appear to arise from term 3, while terms 2 and 4 are insignificant. This term is the product of the interannual windstress anomaly and the climatological velocity.
Figure 2 GFDL Comparison of zonal (black) and meridional (red)
work terms for the GFDL model. A) total term, b) term
2, c) term 3, d) term 4. overlaid in green is Nino3.4
from the model. Black and green have had 12 month running means applied. Note
that all plots have been divided by 1011. SST has not had the
seasonal cycle removed here – need to do this.
Figure 3 Term 1 for a) the zonal windstress and velocity and b) the meridional. Note that a) is larger by a factor of 10.
Understanding the Available Potential Energy
A zero energy state of the ocean would have horizontal isopycnals. If an isopycnal slants upwards then it is storing energy as heavier water is being raised. To calculate the APE then ρ is considered as a perturbation to this horizontal background state.
where is the horizontal reference field found by averaging over the region of study. The remaining part gives the variations from this state including the annual mean state of the sloping thermocline due to annual mean trade winds. It also describes the seasonal cycle and the interannual variability.
The APE is calculated from
According to Oort et al. 1989, with from the background density field described above. Goddard and Philander convert this equation into depth displacement h for a shallow water model. We leave it in terms of ρ.
Density is then partitioned into the seasonal and interannual components . Note that the seasonal cycle contains the annual mean. The APE can then be written
(1) (2) (3)
Term (1) relates to the climatology and (2) and (3) to ENSO. The three terms are plotted with Nino3.4 SST below. The first term is relatively constant compared to the other two with values between 2.1 and 2.5 *1018. It is also always positive representing the constant slope of the thermocline forced by the trade winds. The third term is the perturbation from this slope and is squared so it is always positive. It becomes large during El Nino events but remains small an positive in La Nina events
Figure 4 A comparison of terms
that make up the APE. A) Climatology (term 1), b) term (2), c) term (3) and d)
term (2) + term (3). The SST anom is overlaid in red.
Study_energy.jnl