Depth decay rate for surface gravity wave pressure and velocity

Linear governing equations are formulated for the depth decay of the pressure and velocity variations associated with propagating surface gravity waves. These governing equations come from combining Bernoulli’s equation for steady frictionless flow along a streamline and the crossstream force balance involving gravity, the centrifugal force and a pressure gradient. Qualitative solutions show that the pressure decreases downward faster than the velocity does and at a rate that is probably not the normal exponential decrease, which does not agree with the classical result. The radius of curvature of the streamlines is a non-constant coefficient in these equations and it needs to be supplied, either from measurements or another theory, in order to complete the solution of the derived governing equations. There is no sensitivity of the solution to the exact path the radius of curvature takes between its minimum value at the surface of a crest and trough and infinity at great depth. In the future measurements, perhaps streak photographs, will be needed to distinguish between the new and old theories.

SW Origin of North Pacific’s Wide Warm Surface Current

There is a long and wide continuous trough of deep mixed layers connecting the tropical western North Pacific Ocean with the offshore waters of the coast of California. Relatively warm water that is nearly uniform vertically fills the trough, which is concluded here to be a northeastward flow joining the wide warm surface current at mid-latitudes off California documented earlier. Evi-dence for the trough comes from a North Pacific atlas based on very many indi-vidual mixed layer depth data points, taken over a 27-year period, compiled (av-eraged) in monthly mean charts with contours of constant mixed layer depth dis-played. BTs (bathythermographs) were used to record temperature versus depth continuously from which the mixed layer depths were determined. Centerline curves, connecting the deepest mixed layer depths, which approximate the mid-dle of the troughs, are constructed from the atlas and are presented for all twelve months. In going from west to east, these curves bend counterclockwise, gradu-ally most of the way then more markedly near California. The curves for the summer months come closer to California than any of the other ones do, suggest-ing that the warm current itself is nearest to California in summer. Confirmation of the prediction awaits future efforts.

North Pacific Month to Month SST Changes

Month to month changes in the SST of the North Pacific, on the eastern side at mid-latitudes, are studied based on 30 years of ship-injection temperatures. Along both 40 and 35 N the SST maximum shifts west in summer, but it starts west at 35 N two months sooner than at 40 N. In July the maximum at 40 N is at the same location as the maximum at 35 N was in June: 155 W. Since the longitudinal SST maximum in the eastern North Pacific has previously been identified as the signature of a very wide, warm and sluggish current permanently flowing northeast off California, the month to month SST changes are used to estimate its mean speed: 10 - 20 cm/sec. Also the month to month SST changes indicate that in summer a new body of warm water goes north, in a pulse- like movement, to the west of the existing wide warm current. This is consistent with the need of the western equatorial ocean to export more heat northward out of the tropics in summer due to the increased absorption of solar radiation in the surface layer in that season.

Seasonal sea surface temperatures of the North Pacific

Mean seasonal surface temperatures of the North Pacific are illustrated in three maps. Twenty nine years of ship-injection temperatures are used for the whole North Pacific (north of 20?N). Map number two shows geographical regions of the month of highest sea surface temperature. There are two broad bands in the central and eastern basin, trending northeast/southwest, such that the September band lies east of the August band along a given latitude line. Map three depicts regions of the lowest monthly mean temperatures. March is the most common month, but in the middle of the ocean is a band of Februarys trending northeast/southwest. These features on maps two and three are interpreted in terms of the newly proposed wide warm surface current and its seasonal variations, mainly in horizontal position, flowing northeastward off California. It has not been found possible to compare maps two and three with the results from any earlier work. Map one shows the mean seasonal range of surface temperature, which has a character similar to maps going all the way back to the late 1800s, but is based on considerably more data.

Lift Force on a Circular Arc Wing

The lift force is calculated for a gliding wing with a circular arc top and a flat bottom in a uniform fluid. It is: constρU2/R0, where??is the constant fluid density, U is the uniform flow speed far from the wing and??is the radius of curvature of the wing’s top surface. To obtain this result two non-linear differential equations in pressure and velocity are combined into one linear governing equation for velocity, which contains a non-constant coefficient, R(z), the radius of curvature of the streamlines above the wing as a function of the vertical coordinate z. Bernoulli’s principle along a streamline?and the force balance across a streamline (pressure gradient equals centrifugal force) are the starting equations. A solution to the governing equation is derived by providing an algebraic function for R(z)?that is consistent with observations, and the order of magnitude one constant?in the lift force is worked out. It is believed that the present approach to understanding the lift force on a wing has not been tried before. More theoretical and observational work are needed to better specify R(z).

Southward surface flow in the central South Pacific

A large-scale surface flow with a southward component is proposed for the central South Pacific Ocean based on an interpretation of existing closely spaced and accurately measured temperatures and salinities along two latitudes in two different southern hemisphere winters: 28o S (Scorpio) and five degrees south of that (WOCE). Such a southward flow is not predicted from theory nor is it shown on current charts and globes. The observed longitudinal maximum in surface temperature along 28o S is centered around 130o W and has an amplitude of at least 5o C and an east/west range of about 60o of longitude. This striking feature is most easily explained by horizontal transport from latitudes closer to the equator. Since temperature atlases show that equatorial surface temperatures are always highest in the west, the origin of the warm water probably is toward the western side of the ocean as well. Thus the surface flow surrounding the longitudinal temperature maximum should be directed to the southeast. Where the surface temperatures are maximum the mixed layer depths are relatively large in a convex downward lens with maximum depths of 100 m;a correlation that is consistent with warm water moving south and being cooled from above. Salinities are maximum near the temperature maximum, also suggesting that the source of the surface flow is at low latitudes, where evaporation is usually expected to exceed precipitation. It is conjectured that the large-scale southeastward flow of the South Pacific is the analogue of the northeastward wide warm current off California documented over 30 years ago.

Cone Rotating in a Fluid

When a solid cone with smooth side and base rotates about its long axis in a still fluid, theory says that the cone will advance along the direction of the axis, base first and apex last. Bernoulli’s law for closed streamline loops is combined with the cross-stream force balance between the centrifugal force and a pressure gradient in order to obtain the result, which is believed to be new. Confirmation of the prediction awaits observational evidence.

Bernoulli Weather or Not?

Thirty five day records of wind speed and atmospheric pressure, measured and recorded every two hours, were obtained on a ship that sailed from California to Japan along latitude 35 N in the spring of 1976. Time variations of two days, which can be seen in the raw data of both variables, were brought out more clearly by a two-step smoothing process followed by a subtraction to reduce both the lower and higher frequencies. Comparing the two similarly processed records shows that when the wind speed is relatively strong, the pressure is relatively weak, and vice versa, at the two day time scale. An argument is given that Bernoulli’s law was operating in those weather systems.

Double North Pacific High in Summer

An example of sea level pressure (SLP) and sea surface temperature (SST) is displayed for a summer month based on historical monthly mean data for the North Pacific. A double North Pacific High (NPH) co-occurred with a double large-scale SST maximum along 40 N. Centers of the two NPHs had very nearly the same longitudes as did the SST maxima. Seven similar coincidences happened within the 30-year records. These particular associations between extrema of SLPs and SSTs enhance a previously published conjecture that single and double NPHs are caused by heat transfer from the sea surface to the atmosphere. The eastern SST maximum is the signature of a permanent wide warm surface current flowing northeast off California. To the west of it in the summer is a transient wide warm surge of surface water flowing north as it crosses mid-latitudes. These are the heat sources that generate the single and double NPHs.

Lift on a Low Speed Circular Arc Wing due to Air Compression

A fluid flow model consisting of Bernoulli’s law in its normal form, the equation of state of air, and the cross-stream force balance between a downward pressure gradient and the upward centrifugal force on fluid particles moving along curved streamlines over the top circular wing surface involving three equations in three unknowns (pressure, density and velocity) are solved to show that both density and pressure decrease upward as the inverse square of the distance from the circle’s center, and the velocity is independent of that dis-tance. These derived characteristics are used to explain the lift force on the wing in what is believed to be a novel way.

Centrifugal force: an appreciation

The centrifugal force is used to increase the physical understanding of five examples taken from fluid dynamics, geophysics and the solar system, as well as four hypothetical orbital problems. Each example involves a balance of forces between the centrifugal force and one or two other forces, such as a pressure gradient and a component of the force of gravity. Among the examples chosen for examination are: the orbital motion of fluid particles in surface grav-ity waves, the boundary layer character of steady flow next to a curved rigid surface, the tornado, the rotating self-gravitating mass and the three-body problem.

Large Volume of Warm Water outside the Gulf Stream

Mixed layer depths are presented for the mid-latitude North Atlantic obtained from BT (bathythermograph) measurements for a 40 degree longitude band starting at Africa and moving west along 30 N. During February, 1959, 250 BTs were made in this region and they all show a distinct mixed layer depth greater than or equal to 85 meters. By comparing this east/west vertical temperature section with two other BT sections, one along 16 N and the other along 40 N taken one year earlier, but also in the cooling season (October, November), it is proposed that there was a northward drift in the surface layer between 16 N and 40 N that was cooled from above. Such a wide poleward flow of warm water, outside the Gulf Stream, is suggested to be the analogue of the permanent wide warm current off California in the North Pacific studied in some detail earlier.

Henderson Island’s Plastic: An Explanation

Plastic has been accumulating on the beaches of Henderson Island in vast quantities according to recent news reports. It is proposed that the plastic is brought to the island by a very broad permanent surface current flowing southeastward past the island. Other characteristics of the flow are that its temperature is relatively high, its depth is shallow (about 100 m), its speed is sluggish (10 - 20 cm/sec), and by broad is meant more than 5000 km along 28 S. Henderson Island is located at the east/west midpoint of this wide warm current (130 W). By knowing more definitely where the plastic is coming from, than the vague suggestions provided by the news sources, it may be possible in the future to slow down or stop the piling up of trash on what were pristine beaches of this World Heritage Site.

Pacific Tropics Are Warmer than Atlantic

Pacific tropics are warmer than Atlantic tropics. This can be seen immediately from a world’s sea surface temperature atlas, but readings from it of the latitudinal bandwidths between 80 F isotherms in the open waters of both oceans make this result very clear. Explanations are offered by introducing what has already been proposed for the large-scale surface circulations of the Pacific and incorporating conjectures for the circulation of the upper layer of the South Atlantic.

Second Summer in NW Indian Ocean

Two independent SST atlases have confirmed that there is a double seasonal signal in the surface layer of the northwestern Indian Ocean. The area enclosed by the 80F isotherm increases from January to a maximum in May followed by a decrease, a cool-down, to August, which has been explained recently. Then there is a second maximum of warm surface area in October/November, called a “second summer” for convenience. A hypothesis is proposed to explain this unusual second summer feature by extrapolating from available data. During July, August and September, the sea level slopes downward from the equator to the north providing a horizontal force to drive the warm surface water accumulated in the equatorial region back into the NW Indian Ocean basin. New observations are needed to check up on the hypothesis.

Bernoulli Loops in Smoke Rings

Bernoulli’s law is applied to the closed streamlines of a smoke ring, and the centrifugal force of the curved flow is balanced by a pressure gradient. Two equations in two unknowns, pressure and velocity, are combined into one equation in one unknown, velocity. Solving the governing equation algebraically produces a radial shear in the velocity such that the speed decreases outward inversely as the radius increases, which is the main result. Measurements are needed to verify the predicted structure of the velocity field.

Downwelling by Surface Gravity Waves?

Based on observational evidence and the known physical characteristics of surface gravity waves, an argument is made that downwelling is not a significant feature over the life history of these waves under the usual conditions existing in the open oceans. Since it has recently been predicted that upwelling due to surface gravity waves should occur within a storm at sea, when the waves are growing in amplitude, the contrast is explained. As a result the importance of the upwelling concept due to waves in stimulating biological productivity is further emphasized, and the possibility for reducing global warming is worth repeating.

Upwelling by Surface Gravity Waves

Three upwelling mechanisms are compared that involve progressive surface gravity waves. In all cases water is pumped up from the depth of wave influence. Two of the methods that are not fully discussed in print before can occur in nature. During wind generation of surface waves in the open sea wave amplitudes and the Stokes drift increases in the direction of wave propagation implying that the Stokes drift is divergent and requiring, as a consequence, that mass be supplied to the surface from below the wave layer. The depth of wave influence of wind generated swell can reach below the light zone, so nutrients can be brought up into the sunlight and biological activity enhanced and global warming ameliorated. A third method for shallow water is the oscillation of a paddle hinged and fixed to the bottom and moved by surface waves passing by.

South Atlantic’s Surface Circulation: A Note

SSTs of the South Atlantic suggest there is a side by side surface circulation with north-ward cold water to the east and southward warm water to the west. Then logic predicts that both currents should be stronger in the southern hemisphere summer than in winter in order to maintain the heat balance, because the upper 100 m of the water column absorbs more solar energy in summer, especially at lower latitudes. This prediction is confirmed by the seasonal variation of the deflection angle of the sea surface isotherms in the middle of the ocean, measured counterclockwise from the east, which are highest in summer and lowest in winter. It is assumed that the stronger the north/south component of a current is, the greater the deflection is of the isotherm from constant latitude.

NW Indian Ocean’s SSTs Are Bi-Seasonal

Two independent atlases agree that in the northwest Indian Ocean the sea surface temperatures cool down markedly in the spring and also that they have a significant double seasonal cycle, unlike any other ocean in both cases. Horizontal advection is proposed to play an important part in causing these unusual features to occur.