Whitehead John A.

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Whitehead

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John A.

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0000-0003-0910-049X

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The similarity solution for turbulent mixing of two-layer stratified fluid

(Springer, 2008-06-19) Whitehead, John A.

Experiments are reviewed in which a two-layer salt-stratified tank of water was mixed by turbulence. The density profile began as a single step and evolved to a smooth mixed profile. The turbulence was generated by many excursions of a horizontally moving vertical rod with Richardson number Ri > 0.9 and Reynolds Number Re > 600. There was almost perfect collapse of all the profiles to one universal profile as a function of a similarity variable. We develop a theoretical model for a simple mixing law with a buoyancy flux that is a function of internal Richardson number Rii. A similarity equation is found. A flux law that increases with small Rii and decreases with large Rii is considered next. Since no analytical solution is known, the similarity concept is tested by numerically integrating the equations in space and time. With buoyancy flux monotonically increasing with internal Richardson number, the similarity approach is valid for a profile starting from a slightly smoothed step. However, a shock forms for a mixing law with higher initial Rii (so that buoyancy flux decreases with Richardson number) and the similarity approach is invalid for those initial conditions.
Turbulent mixing of two-layer stratified fluid

( 2007-11-09) Whitehead, John A. ; Stevenson, Ian

A two-layer salt-stratified tank of water was mixed by turbulence generated by many excursions of a horizontally moving vertical rod. The objective is to observe the timedependent response of the mean density field for ranges of Richardson number Ri>0.9 and Reynolds Number Re>600. As the density profile of the fluid gradually evolves from a single step to a mixed state over a wide range of time, there is almost perfect collapse of all the profiles to one universal profile as a function of a similarity variable. Although the turbulent diffusion is not constant, the value in the limit of small stratification has similar magnitude to values found by others.
Final technical report : contract N00014-81-C-0010 : 1 October 1980 - 30 September 1982

(Woods Hole Oceanographic Institution, 1983-03) Whitehead, John A.

Laboratory, theoretical and numerical research was conducted into the structure and stability of baroclinic non-linear currents in a rotating fluid. A rotating version of the dam-break problem in which a . density current is generated after a barrier has been removed was studied. The speed of the current and its width and depth were measured by Whitehead (1981) and more extensively by Stern, Whitehead, and Lien Hua (1982), who report the experiments and compare the results to theory. Properties of a limiting bore solution for rotation density currents predicted earlier by Stern are incorporated into the above theory to predict the speed of the nose of the current. Experiments are described in which the current width is measured to be in reasonable agreement with the theory. Theoretical studies of the stability of a free isolated baroclinic jet whose free surface in cross-section intersects the water surface at two points by Griffiths, Killworth and Stern (1982) was undertaken. The waves permit the release of both kinetic and potential energy. They can have rapid growth rates, thee-folding time for waves on a current with zero potential vorticity being close to one-half of a rotation period. Experiments with a current of buoyant fluid at the free surface of a lower layer were also conducted. The current was observed to be always unstable. Killworth and Stern (1982) showed that a coastal density current in a rotating system is unstable to downstream wave disturbances when the mean potential vorticity increases towards the (vertically-walled) coast and when the mean current vanishes there. Other new instability modes were also found which do not require the potential vorticity extremum of quasi-geostrophic theory. Paldor, in his Ph.D. thesis, used Rayleigh integral to prove that an unbounded geostrophic front of uniform potential vorticity is stable with respect to small perturbations of arbitrary wavelength. Stern and Paldor (1983) used extremum concepts to analyze large amplitude disturbances in a boundary layer shear flow with an inviscid and longwave theory. It was found that initially weak horizontal convergences were concentrated and amplified in time.
Effect of potential vorticity on flow rate through a gap

(American Geophysical Union, 2005-07-16) Whitehead, John A.

Mathematical solutions for constant potential vorticity critically controlled flow through ocean passages are complicated and not available in simple form. Therefore, to provide formulas for numerical circulation and ocean climate models, two simple formulas for volume flux are developed here. They are fitted to numerical values of the critical flux for constant potential vorticity flow over a flat bottom through a constriction. The two formulas of increasing complexity agree with the numerical values to better than 6% and 1.4%. These flux values are up to 24% less than the values of flux from zero potential vorticity formulas presently applied to ocean passages. The most precise new formula is used to predict flux magnitude through nine ocean passages that have current meter measurements. The size of the revisions compared to zero potential vorticity predictions is a few percent in the direction of better agreement. For further improvement between prediction and observation, other factors such as realistic bottom topography, friction, mixing, waves, and eddies must be included.
Instability and freezing in a solidifying melt conduit

( 2010-10-23) Holmes-Cerfon, Miranda C. ; Whitehead, John A.

Previous works have shown that when liquid flows in a pipe whose boundary temperature is below freezing, a tubular drainage conduit forms surrounded by solidified material that freezes shut under the appropriate combination of forcing conditions. We conduct laboratory experiments with wax in which the tube freezes shut below a certain value of flux from a pump. As the flux is gradually decreased to this value, the total pressure drop across the length of the tube first decreases to a minimum value and then rises before freezing. Previous theoretical models of a tube driven by a constant pressure drop suggest that once the pressure minimum is reached, the states for a lower flux should be unstable and the tube should therefore freeze up. In our experiments, flux and pressure drop were coupled, and this motivates us to extend the theory for low-Reynolds number flow through a tube with solidification to incorporate a simple pressure drop-flux relationship. Our model predicts a steady-state relationship between flux and pressure drop that has a minimum of the pressure as the flux is varied. The stability properties of these steady states depend on the boundary conditions: for a fixed flux, they are all stable, whereas for fixed pressure drop, only those with a flux larger than that at the pressure drop minimum are stable. For a mixed pressure-flux condition, the stability threshold of the steady states lies between these two end members. This provides a possible mechanism for the experimental observations.
Temperature and velocity measurements of a rising thermal plume

(John Wiley & Sons, 2015-03-04) Cagney, Neil ; Newsome, William H. ; Lithgow-Bertelloni, Carolina ; Cotel, Aline ; Hart, Stanley R. ; Whitehead, John A.

The three-dimensional velocity and temperature fields surrounding an isolated thermal plume in a fluid with temperature-dependent viscosity are measured using Particle-Image Velocimetry and thermochromatic liquid crystals, respectively. The experimental conditions are relevant to a plume rising through the mantle. It is shown that while the velocity and the isotherm surrounding the plume can be used to visualize the plume, they do not reveal the finer details of its structure. However, by computing the Finite-Time Lyapunov Exponent fields from the velocity measurements, the material lines of the flow can be found, which clearly identify the shape of the plume head and characterize the behavior of the flow along the plume stem. It is shown that the vast majority of the material in the plume head has undergone significant stretching and originates from a wide region very low in the fluid domain, which is proposed as a contributing factor to the small-scale isotopic variability observed in ocean-island basalt regions. Lastly, the Finite-Time Lyapunov Exponent fields are used to calculate the steady state rise velocity of the thermal plume, which is found to scale linearly with the Rayleigh number, in contrast to some previous work. The possible cause and the significance of these conflicting results are discussed, and it is suggested that the scaling relationship may be affected by the temperature-dependence of the fluid viscosity in the current work.
A laboratory model of a cooled continental shelf

(Woods Hole Oceanographic Institution, 1993-06) Whitehead, John A. ; Frazel, Robert E.

A laboratory model of wintertme cooling over a continental shelf has a water surface cooled by air in an annular rotating tank. A flat shallow outer "continental shelf" region is next to a conical "contiental slope" bottom and a flat "deep ocean" center. The shelf flow consists of cellular convection cells descending into a region with very complicated baroclinic eddies. Extremely pronounced fronts are found at the shelf break and over the slope. Associated with these are sizable geostrophic currents along the shelf and over shelf break contour. Eddies are particularly energetic there. Cooling rate is compared with temperature difference between "continental shelf" and "deep ocean". Scaling considerations produce an empirical best fit formula for temperature difference as a function of cooling rate. This produces a relatively straight regression line over a wide range of rotation rates, shelf depths and cooling rates. If this formula is valid for the ocean, water over continental shelves will be much colder due to constraints imposed by rotation of the earth than if the fluid were not rotating.
Deep-water flow over the Lomonosov Ridge in the Arctic Ocean

(American Meteorological Society, 2005-08) Timmermans, Mary-Louise ; Winsor, Peter ; Whitehead, John A.

The Arctic Ocean likely impacts global climate through its effect on the rate of deep-water formation and the subsequent influence on global thermohaline circulation. Here, the renewal of the deep waters in the isolated Canadian Basin is quanitified. Using hydraulic theory and hydrographic observations, the authors calculate the magnitude of this renewal where circumstances have thus far prevented direct measurements. A volume flow rate of Q = 0.25 ± 0.15 Sv (Sv ≡ 106 m3s−1) from the Eurasian Basin to the Canadian Basin via a deep gap in the dividing Lomonosov Ridge is estimated. Deep-water renewal time estimates based on this flow are consistent with 14C isolation ages. The flow is sufficiently large that it has a greater impact on the Canadian Basin deep water than either the geothermal heat flux or diffusive fluxes at the deep-water boundaries.
Abrupt transitions and hysteresis in thermohaline laboratory models

(American Meteorological Society, 2009-05) Whitehead, John A.

As a driving parameter is slowly altered, thermohaline ocean circulation models show either a smooth evolution of a mode of flow or an abrupt transition of temperature and salinity fields from one mode to another. An abrupt transition might occur at one value or over a range of the driving parameter. The latter has hysteresis because the mode in this range depends on the history of the driving parameter. Although assorted ocean circulation models exhibit abrupt transitions, such transitions have not been directly observed in the ocean. Therefore, laboratory experiments have been conducted to seek and observe actual (physical) abrupt thermohaline transitions. An experiment closely duplicating Stommel’s box model possessed abrupt transitions in temperature and salinity with distinct hysteresis. Two subsequent experiments with more latitude for internal circulation in the containers possessed abrupt transitions over a much smaller range of hysteresis. Therefore, a new experiment with even more latitude for internal circulation was designed and conducted. A large tank of constantly renewed freshwater at room temperature had a smaller cavity in the bottom heated from below with saltwater steadily pumped in. The cavity had either a salt mode, consisting of the cavity filled with heated salty water with an interface at the cavity top, or a temperature mode, in which the heat and saltwater were removed from the cavity by convection. There was no measurable hysteresis between the two modes. Possible reasons for such small hysteresis are discussed.
Conceptual models of the climate : 2002 program of study, Bounds on turbulent transport

(Woods Hole Oceanographic Institution, 2003-07) Whitehead, John A. ; Busse, Friedrich ; Howard, Louis ; Doering, Charles ; Constantin, Peter ; Caulfield, Colm-cille ; Kerswell, Richard

The subject of "Bounds of Turbulent Transport" was introduced in a series of ten lectures. The six lecturers constitute almost all the contributors to this subject. The subject was introduced and foundations laid by five lectures by F. H. Busse. In the middle of the first week, L. Howard reviewed his historical first approach to this subject and described more recent advances. Additional lectures by P. Constantine, R. Kerswell, C. Caulfield and C. Doering provided modern advances. We trust that the lecture notes will constitute a timely review of this promising subject.
2007 program of studies : boundary layers

(Woods Hole Oceanographic Institution, 2008-06) Cenedese, Claudia ; Whitehead, John A. ; Pedlosky, Joseph ; Lentz, Steven J.

The topic of the Principal Lectures for the forty-ninth year of the program was “Boundary Layers”. The subject centers around those problems in which the boundary conditions lead to a large gradient near the boundary. Nine of this year’s principal lectures were given by Joe Pedlosky and the tenth was given by Steve Lentz. The fluid mechanics of boundary layers was reviewed, first starting from its classical roots and then extending the concepts to the sides, bottoms, and tops of the oceans. During week four, a mini-symposium on “Ocean Bottom and Surface Boundary Layers” gathered a number of oceanographers and meteorologists together to report recent advances. And, finally, Kerry Emanuel of MIT delivered the Sears Public Lecture to a packed hall in Clark 507. The title was “Divine Wind: The History and Sciences of Hurricanes.”
Laboratory studies of stratified convection with multiple states

( 2005-01-14) Whitehead, John A. ; Bradley, Keith F.

A simplified box model of the cooling of a salt-stratified ocean has been constructed in the laboratory to test a theory that predicts multiple equilibria if certain conditions exist. An isothermal basin of water had a thin layer of fresh water over salt water. Beside this was a smaller basin connected to the large basin by horizontal tubes at the top, middle and bottom. The small basin was cooled from above. If the top tube has more flow resistance than the bottom tubes, theory indicates that as cooling temperature T* is made colder, there is a sudden transition between two flow states. The velocities in the tubes jump to greater values, while salinity and temperature in the small basin jump to another value. These multiple states are found in the laboratory experiments along with some states that oscillate. Laboratory measurements and layered model calculations for hysteresis and the jump of temperature and salinity agree qualitatively, but there is only rough quantitative agreement.
Rotating hydraulic control : 1997 summer study program in geophysical fluid dynamics

(Woods Hole Oceanographic Institution, 1998-07) Whitehead, John A.

Rotating Hydraulic Control was the topic of the thirty ninth year of the Geophysical Fluid Dynamics (GFD) program at the Woods Hole Oceanographic Institution. This theme was principally centered about those nonlinear problems in which either a free surface or internal stratification is so modified by flow that it acts to choke off increased flux as the forcing is increased. It is a peculiar form of convection, which shares many constraints with more general buoyancy driven motion but which has its own internal limits. Lectures and seminars were given by GFD staff and visitors, most of whom are founders of this young field of study. This volume contains notes from the talks given by the principal lecturers and written reports on the research projects cared out by the ten student fellows. The volume, therefore, summarizes a sizable percentage of the present understanding of the topic of Rotating Hydraulic Control.
Deepwater overflow through Luzon Strait

(American Geophysical Union, 2006-01-10) Qu, Tangdong ; Girton, James B. ; Whitehead, John A.

This study examines water property distributions in the deep South China Sea and adjoining Pacific Ocean using all available hydrographic data. Our analysis reveals that below about 1500 m there is a persistent baroclinic pressure gradient driving flow from the Pacific into the South China Sea through Luzon Strait. Applying hydraulic theory with assumptions of zero potential vorticity and flat bottom to the Luzon Strait yields a transport estimate of 2.5 Sv (1 Sv=106 m3 s-1). Some implications of this result include: (i) a residence time of less than 30 years in the deep South China Sea, (ii) a mean diapycnal diffusivity as large as 10-3 m2 s-1, and (iii) an abyssal upwelling rate of about 3×10-6 m s-1. These quantities are consistent with residence times based on oxygen consumption rates. The fact that all of the inflowing water must warm up before leaving the basin implies that this marginal sea contributes to the water mass transformations that drive the meridional overturning circulation in the North Pacific. Density distributions within the South China Sea basin suggest a cyclonic deep boundary current system, as might be expected for an overflow-driven abyssal circulation.
Fluid flow with three upstream configurations in freezing tubes

(American Geophysical Union, 2021-05-17) Whitehead, John A.

The accumulation of frozen liquid around a central passageway of melt as it flows through a freezing region can make calculations very challenging. To both illustrate and to quantify some of these challenges from freezing, a model equation is developed. It simplifies the solution of Holmes (2007, https://gfd.whoi.edu/wp-content/uploads/sites/18/2018/03/MHolmesGFDReport_30151.pdf) for low Reynolds number single component liquid flow through a long tube that has a wall kept at subfreezing temperature. This model equation is used in conjunction with three different upstream configurations, each with parameters expressing their behavior. Analytical and numerical results give the parameters that have criteria for: the freezing of a compressible upstream reservoir that includes oscillatory behavior; the freezing of flow fed through a constriction with a large upstream pressure, just like a dripping water faucet during winter; the evolution of flow in multiple tubes connected by an upstream manifold, where some tubes end up with full flow and others freeze shut. Numerical runs with 1,000 tubes give a formula for the spacing between actively flowing (non-frozen) tubes over wide ranges of the two upstream parameters (flow rate and manifold resistance). Results have implications in various areas in earth science. Some are: oscillatory and freezing shut criteria for flow of magma from a compressible region, a criterion for wintertime ice accumulation at natural springs, and the spacing between volcanos.
Dimensions of continents and oceans – water has carved a perfect cistern

( 2017-03-26) Whitehead, John A.

The ocean basins have almost exactly the correct surface area and average depth to hold Earth’s water. This study asserts that three processes are responsible for this. First, the crust is thickened by lateral compression from mountain formation. Second, Earth’s continental crust is leveled by erosion. Third, due to the efficiency of erosion, the average elevation is a few hundred meters above sea level. A theoretical fluid model, suggested partly by laboratory experiments, includes an ocean of specified depth. The resulting continents are tabular (that is, their elevation view is rectangular). The surface lies above sea level, contributing to a well-known double maximum in Earth’s elevation corresponding to continents and ocean basins. Next, a simple hydrostatic balance between continent and ocean gives average depth and area of present oceans and continents within 33%. Further calculations with a suitable correction to fit present Earth cover a wide range of possible crust volumes for earlier Earth. With the present water volume, ocean area always exceeds 25% of the globe. For all possible water volumes, average continental crust thickness always exceeds 23.4 km. This may explain why cratons have thicknesses comparable to younger crust so that they are found on Earth’s surface today. Therefore, mountain building, and erosion have enabled water to carve its own cistern in the form of the accumulated ocean basins. The wide range of areas and depths of oceans and continents found here can constrain models of early earth. Similar calculations can be done for earthlike planets as well.
Trans-equatorial bottom water flow in the western Atlantic : volume XLVI in a series of reports presenting data from moored current meters

(Woods Hole Oceanographic Institution, 1997-02) Tarbell, Susan A. ; Whitehead, John A. ; Hall, Melinda M. ; McCartney, Michael S.

Current and temperature measurements from Vector Averaging Current Meters (VACMs) deployed from September 1992 to June 1994 as part of the Deep Basin Experiment (DBE) measuring the trans-equatorial water flow are presented. Salinity and temperature measurements from Conductivity/Temperature/Depth (CTD) casts taken during the mooring deployment and recovery cruises are also presented. Six mooring sites were occupied with a total of 24 vector averaging current meters and 4 Aanderaa current meters. Three nominal depths (3900, 4100 and 4300 m.) were occupied on each mooring. Three of the 6 moorings had current meters at additional depths. Basic data from the vector averaging current meters are presented both in statistical tables and graphically as histograms, scatter plots, progressive vector diagrams and spectral diagrams. One day Gaussian filtered plots are shown in composite displays of variables versus time. Temperature and salinity profies and e/s plots for 22 CTD stations are presented.
Convection driven by temperature and composition flux with the same diffusivity

( 2017-05) Whitehead, John A.

Temperature, pressure, and composition determine density of fluids within the earth, the ocean, our atmosphere, stars and planets. In some cases, variation of composition component C competes equally with temperature T to determine buoyancy-driven flow. Properties of two-dimensional cellular convection are calculated with density difference between top and bottom boundaries determined by difference of temperature T (Dirichlet boundary conditions, quantified by Rayleigh number Ra that is positive destabilising), fluxes of C (Neumann boundary conditions quantified by Raf that is positive stabilising), and Prandtl number Pr. Numerical solutions in a 2-dimensional rectangular chamber are analysed for Prandtl numbers Pr=1,∞. For Ra and Raf>0 and Raf above approximately 300, subcritical instability separates T-driven convection from C-dominated stagnation. The flow is steady but a sudden change in Ra or Raf produces decaying pulsations to the new flow. A boundary layer solution for rapid flow exists in which T, which has the Dirichlet condition, is more sensitive to flow speed than C with the Neumann condition. A new type of pulsating flow occurs for Ra and Raf<0. The pulsations are characterised by slow flow with gradually strengthening compositional plumes in a thermally stratified flow interrupted by rapid flow with gradually weakening compositional plumes. In this slow speed range, C is more sensitive to speed than T.
A laboratory model of vertical ocean circulation driven by mixing

(American Meteorological Society, 2008-05) Whitehead, John A. ; Wang, Wei

A model of deep ocean circulation driven by turbulent mixing is produced in a long, rectangular laboratory tank. The salinity difference is substituted for the thermal difference between tropical and polar regions. Freshwater gently flows in at the top of one end, dense water enters at the same rate at the top of the other end, and an overflow in the middle removes the same amount of surface water as is pumped in. Mixing is provided by a rod extending from top to bottom of the tank and traveling back and forth at constant speed with Reynolds numbers >500. A stratified upper layer (“thermocline”) deepens from the mixing and spreads across the entire tank. Simultaneously, a turbulent plume (“deep ocean overflow”) from a dense-water source descends through the layer and supplies bottom water, which spreads over the entire tank floor and rises into the upper layer to arrest the upper-layer deepening. Data are taken over a wide range of parameters and compared to scaling theory, energetic considerations, and simple models of turbulently mixed fluid. There is approximate agreement with a simple theory for Reynolds number >1000 in experiments with a tank depth less than the thermocline depth. A simple argument shows that mixing and plume potential energy flux rates are equal in magnitude, and it is suggested that the same is approximately true for the ocean.
Constraining the source of mantle plumes

( 2016-01-08) Cagney, Neil ; Crameri, Fabio ; Newsome, William H. ; Lithgow-Bertelloni, Carolina ; Cotel, Aline ; Hart, Stanley R. ; Whitehead, John A.

In order to link the geochemical signature of hot spot basalts to Earth’s deep interior, it is first necessary to understand how plumes sample different regions of the mantle. Here, we investigate the relative amounts of deep and shallow mantle material that are entrained by an ascending plume and constrain its source region. The plumes are generated in a viscous syrup using an isolated heater for a range of Rayleigh numbers. The velocity fields are measured using stereoscopic Particle-Image Velocimetry, and the concept of the ‘vortex ring bubble’ is used to provide an objective definition of the plume geometry. Using this plume geometry, the plume composition can be analysed in terms of the proportion of material that has been entrained from different depths. We show that the plume composition can be well described using a simple empirical relationship, which depends only on a single parameter, the sampling coefficient, Sc. High-Sc plumes are composed of material which originated from very deep in the fluid domain, while low-Sc plumes contain material entrained from a range of depths. The analysis is also used to show that the geometry of the plume can be described using a similarity solution, in agreement with previous studies. Finally, numerical simulations are used to vary both the Rayleigh number and viscosity contrast independently. The simulations allow us to predict the value of the sampling coefficient for mantle plumes; we find that as a plume reaches the lithosphere, 90% of its composition has been derived from the lowermost 260−750 km in the mantle, and negligible amounts are derived from the shallow half of the lower mantle. This result implies that isotope geochemistry cannot provide direct information about this un-sampled region, and that the various known geochemical reservoirs must lie in the deepest few hundred kilometres of the mantle.

Whitehead John A.

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