Turbulence in the Atmosphere and Oceans
Martin I. Hoffert
New York University
New York, New York
Turbulence in the Atmosphere and Oceans
Martin I. Hoffert
New York University
New York, New York
Hoffert discussed issues of scale with regard to turbulence in the oceans and atmosphere. Turbulence is an order of magnitude subject; there is no rigorous theory of turbulence. From a climate perspective, we are challenged by an atmosphere, oceans and Earth that are highly anisotropic (having different properties when measured along different axes or directions), with a ratio of 10 million to 1 in characteristic land scales. The horizontal and vertical dimensions and not comparable and fluctuations in all the fields are highly anisotropic in the large scale circulation that determines climate. How does turbulence enter into the large scale prediction of global climate? What problems exist and what can be elucidated through a better understanding of turbulence?
Very small scale turbulence (eddy scale size on the order of 1 meter to a few meters, out of 4000 meters of ocean depth) is associated with vertical mixing in the oceans. This small scale turbulence actually controls the large scale circulation of the oceans and is simply input into GCMs (not explicitly resolved). Hoffert predicts that it will be many years before we can explicitly resolve this turbulence in computer simulations.
Equator to Pole Gradient
Earth has a poleward flow of heat carried by both the atmosphere and oceans; the equator would be far warmer and the poles far colder if this flow did not take place. This gradient is important to Earth's climate. As the climate changes, what happens to this gradient? The climate sensitivity question is a zeroth order problem: how much temperature change will be brought about by a doubling of carbon dioxide concentration in the atmosphere? The first order problem then is what happens to the equator to pole temperature gradient? How will the climate change at different latitudes?
Hoffert makes the case that the poleward heat flow is not being properly represented in GCMs; nor do current GCMs properly represent what we know from paleoclimate data: that tropical temperatures remain relatively stable while polar temperatures change much more dramatically.
Small scale turbulence actually controls the large scale circulation of the oceans and is simply input into GCMs (not explicitly resolved).
Figure 1.16, for example, illustrates the equator-to-pole surface temperature difference relative to today's for two very different climates: (i) the middle Cretaceous 100 million years before present (100 MY BP) and (ii) the Last Glacial Maximum (LGM) 18 thousand years before present (18 kY BP). Note that temperature changes are relatively small near the equator, but large near the poles. This implies a much flatter equator-to-pole temperature distribution during hot periods like the Cretaceous, and a more non-uniform distribution during cold spells like the LGM. This is a paradox for the theory of eddy transport because large temperature gradients are required by present theories to produce large poleward heat transports in the atmosphere and oceans. How therefore can a flat temperature gradient produce the large heat transport needed to create the "flatness?" The equator-to-pole temperature gradient is now about 45°C; it was less than half of this, about 20°C in the Cretaceous period of 100 million years ago.
Poleward heat flow is not being properly represented in GCMs.
Figure 1.16
Equator-to-pole temperature response 100 Myr and 18 kyr BP. The stippled region is the range of Cretaceous climate response obtained by subtracting the present zonal mean temperature distribution from the 100 Myr BP temperature reconstructions in Thompson and Barron and averaging the two hemispheres. The LGM response is synthesized from CLIMAP sea surface temperature changes averaged for the hemispheres equatorward of sea ice every ten degrees latitude (f < 50o) denoted by and a high-latitude point from the Vostok ice core deuterium isotope record representing the air temperature change over the Antarctic plateau 18 kyr BP. The solid curves are zonal temperature responses of the mid-Cretaceous and LGM.
Another problem is the current parameterizations of turbulence in the interior of the ocean. What are the principle axes of turbulence and should we allow turbulence to vary with stratification in models? And where does the turbulence come from? Diffusivity is 50 times higher than would be expected given the properties of sea water. Why is this the case in the interior of the ocean? The driving force of turbulence is shear flows a strong vertical gradient in current which can be caused by breaking internal gravity waves. Turbulence can also be caused by an instability associated with density. Any small disturbance can cause what is called "convective overturning." Unstable potential temperature causes the flow to become violently turbulent to erase the gradient that created the turbulence. This is because a high density blob of fluid sitting above a low density blob in a gravitational field is unstable. This is called convective adjustment; it exists in both the oceans and atmosphere, and is built into current GCMs.
Turbulent eddies are regions where properties like temperature and salinity fluctuate relative to background currents. They are important in ocean dynamics because heat and tracers like carbon can be transported from one part of the ocean to another for example, from the surface to the deep ocean by turbulence. And they are not generally resolved in ocean models so they have to be represented by approximate equations (parameterizations).
In stably stratified situations (denser water below less dense water), like the ocean thermocline, turbulence comes from the breaking of internal gravity waves - waves associated with the stratification of the oceans and characterized by buoyancy frequency . These gravity waves are ubiquitous in the oceans and they come from different sources. Some may come from surface processes, but the turbulence of the boundary layer does not penetrate very deeply into the ocean. The ocean overturning time is about a thousand years so it takes hundreds of years for surface turbulence to reach the interior of the ocean. But turbulent eddies in the interior ocean decay on time scales on the order of a month. So something locally is creating the vertical turbulence. It is these internal gravity waves that locally produce shear which causes episodic turbulence.
The thermocline (the first 500 meters of the ocean) is a region of the ocean where the temperature drops off relatively rapidly. The depth scale of the thermocline is controlled by the ratio of the vertical diffusivity to the mean upwelling rate of the ocean, and the upwelling itself depends on the vertical diffusivity. It is possible to write an equation for upwelling; the driving force of large scale overturning is related to the equator to pole temperature gradient and to some extent to the salinity gradient. All of this, Hoffert says, is the result of turbulence taking place on a spatial scale of a few meters which is not explicitly resolved in current GCMs. The poleward heat flow can be changed by an order of magnitude depending on this diffusivity.
It takes hundreds of years for surface turbulence to reach the interior of the ocean. But turbulent eddies in the interior ocean decay on time scales on the order of a month.
What causes this interior mixing? Early work by Walter Munk refers to importance of the lunar tides for the spectrum of turbulence in the oceans. The moon pumps the gravity waves which are breaking and creating the turbulence in the ocean, controlling the depth scale of the thermocline, and hence, the ocean circulation. If lunar tides are indeed the source of vertical ocean mixing, it has important implications for the evolution of life on Earth because vertical mixing of nutrients and dissolved oxygen controls the amount of plankton and the rate of photosynthesis in surface waters. The present-day marine biosphere couldn't survive without such mixing and would probably never have evolved in the first place without it. Since life began in the sea, life on Earth without the Moon (and lunar tides) might have evolved along quite different paths, if at all.
Hoffert then focused on the poleward heat flow associated with mean overturning of the thermohaline cells of the ocean which he says is the major component of poleward heat flow in the ocean. In the Cretaceous period, when the thermal gradient was less than half what it is now, the poleward heat transport by eddies was one quarter of its current value, but it should have been more. This is a paradox. The Cretaceous equator-pole temperature gradient can't be reproduced with current GCMs.
Use of Simple Climate Models
Hoffert then discussed the use of simple climate models such as upwelling diffusion models (Harvey et al., 1997). Much policy direction comes from these simple models, perhaps more than the GCMs he says, so they are very important. Most coupled atmosphere/ocean (A/O) GCMs have to use flux adjustments because the models drift from initial conditions when run over long periods of time. This causes some to doubt the models' credibility. Hoffert says that the main advantage of GCMs is that they may eventually be able to help predict the regional distribution of climate change, but until then, simple models may offer more in the way of policy-relevant information.
Data from nine points in time from the Cretaceous through the present demonstrates that the equator to pole temperature distribution flattens as the world gets warmer. Research by Hoffert and colleagues indicates that in order for this flattening to occur, there must also be an increase in poleward heat transport. (When diffusion was held constant in a model, the equator to pole gradient remained at the current value.) It is still an unresolved issue whether and why tropical temperatures remain relatively constant while the higher latitudes are more sensitive. This is a real challenge for turbulence theory.
Hoffert presented results from an upwelling diffusion ocean/climate model for 6 different assumptions about vertical mixing (Figure 1.17).
Much policy direction comes from simple climate models such as upwelling diffusion models, perhaps more than the GCMs.
Figure 1.17
Observed global mean surface temperature variation relative to the year 1890 compared with global warming from 1870-1990 predicted from an upwelling-diffusion ocean/climate model for different World Ocean diffusion laws and Polar Sea mixing and surface warming rates.
These model results show that global warming for greenhouse gases input by human kind for the past hundred years can be significantly different for different models of vertical turbulent mixing which takes place at scales smaller than those which can be explicitly calculated by ocean circulation models.
And Figure 1.18 shows that present day model-derived oceanic heat flows are only about one-sixth of the 6 Petawatt (PW) peak heat flow to the poles implied by satellite data and about one-third of the 3 PW oceanic heat flow we expect as the residual of the total minus atmospheric heat flow. The apparently weak poleward heat flows of modeled oceanic flow components compared with residual implied values has been recognized for some time. It is extremely difficult to raise the poleward heat flow for fixed surface temperature, salinity and stress distributions to 3 or even 2 PW without invoking unrealistically large subgrid scale vertical diffusivity. In other words, the empirical evidence would suggest that the ocean was transporting half of the energy to the poles but none of the models show this.
Figure 1.18
Global meridional heat flows in PW (1 Petawatt = 10 15 W) versus latitude (positive toward the North Pole). The left panel shows total, atmospheric and ocean heat flow derived as a residual by Carissimo et al. The right panel compares the Carissimo residual with ocean poleward heat flows computed by the Semptner-Chervin horizontal eddy-resolving Ocean GCM and Hoffert and colleagues' schematic model.
Conclusions
The way turbulence is represented in current models is not realistic and this has important ramifications for the climate sensitivity predicted by these models. This issue is particularly relevant to issues of scale because processes at very small scales (on the order of 1 meter) ultimately control oceanic overturning and the equator to pole heat flux at the global scale. (There are 6 orders of magnitude difference between them). Roni Avissar says that this is the case in the atmosphere as well. In important ways, the small scale dominates the large scale. There is a fundamental problem in our understanding of the drivers of the global heat flux as our inability to replicate the equator to pole gradient in models shows. There is clearly a problem with accounting for the small scale at the global scale.
Reference
Harvey, Danny, Jonathan Gregory, Martin Hoffert, Atul Jain, Murari Lal, Rik Leemans, Sarah Raper, Tom Wigley, Jan de Wolde, Introduction to Simple Climate Models Used In The IPCC Second Assessment Report, Edited by John T. Houghton, L. Gylvan Meira Filho, David J. Griggs, Kathy Maskell, IPCC Technical Paper II, Intergovernmental Panel on Climate Change, February 1997.
The way turbulence is represented in current models is not realistic and this has important ramifications for the climate sensitivity predicted by these models.
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