My brother owns two commercial roofing companies and tells me that a 'shingle-over' ridge vent does help, but not all by itself. As you've stated, you need some way to create convection of the air currents. Venting through the eves of your house is passive ventilation and doesn't do much to accelerate an air current when the wind's not blowing. Installing whirly bird vents are also subject to outside air movement ... and they're ugly as the dickens, poking out all over the roof.

However--- a gable end vent with a fan controlled by a thermostat can either pull or exhaust a generous amount of air across the attic when the temperature reaches a level you determine. Which way it moves the air depends on either how you install it or if it has a reverse switch. In summer, drawing air into the attic will force hotter air up at the peak to exit through the ridge vent. I suppose there are dozens of other 'solutions' but hope these thoughts give you something to think about.

Common wisdom has ridge/soffit systems being the most effective passive approach. Keeping the attic cooler is secondary to providing ventilation to prevent condensation. I don't know how much difference 10 - 20* makes considering a well installed R30 attic insulation. Nothing beats a gable fan for moving the air. Elk products are second rate in my experience.

Decision has been made. My roofing contractor and I are an the same wavelength. I will have more eyebrow vents than you would believe. They will be as close as possible to the ridge like a few inches. The system will be completely passive. Nothing will wear out. It won't be necessary to crawl into an attic of extremely difficult geometry to install or perhaps service a fan. There will be no fan to vibrate or be heard. The venting at present is so minimal that hot air comes OUT the soffet vents. We don't need a 100% solution just a darn good one. There will be a VAST improvement. All my troubles will be over for ever and the world will be good again.
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I inspected 2 ridge vents and 5 roofing contractors. I found:

The ELK vent has only a 3/4 inch flow passage characteristic dimension. I don't believe that is large enough to establish significant flow for aerodynamic reasons. There are an unbelievable number of zig zag baffles to inhibit the air flow. The allowed opening in the ridge for this contrivance is only 2 inches. Not much is it? Thats a 2 inch inlet to a 3/4 inch high vent passage. Squirrels and birds put stuff in the vent openings especially bluejays. Bugs crawl into em. I looked at another brand which also had the same type drawbacks. Its baffles were foil shaped. Well La T Da. They blocked off 70% of the flow area. Its hot here. The ridge vents get soft and difficult to install; like nailing down spahgetti. They are made from polypropylene which in my experience just does not stand the heat and certainly not the sun.

Insufficient ridge venting cannot be remedied with more vents as a short circuit will result.

Insufficient Eyebrow venting can be remedied with more vents at the same roof level.

Ridge vents might work. Eyebrow vents will work.
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MISLEADING ROOF VENT AREAS

I measured 4 different eyebrow vents, 1 gable shaped vent and eyeballed a bunch on various roofs. I found:

One eyebrow had a screened opening of 3 3/4 X 20 for an area of 75 sq in. But it had a vertical rain guard flap in front and the shortest distance from the edge of that to the lower surface of the xost pasage was 2.25 inch giving a true xost area of 2.25 X 20 = 45 sq in.

The gable shaped vent had 6 louvers at 45º sort of Z shaped each with a rain lip and the top opening was semi-dome shape. The individual screens were about 1.25 inch high but the shortest distance from the rain lip to the lower surface of each louvre was 5/8 inch. The face opening was 8 x 20 overall implying 160 sq in vent. But the actual effective openings including the upper dome totaled 84 sq in.

Great diligence must be exersized in any attempt to match upper roof vent xost areas to soffet inlet vent areas. The nominal dimensions of the various vents may imply xost flow areas at a wide variance from and greater than the actual flow areas.
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fwiw
Part of an old job was energy savings and included some fluid dynamics & calculations for attic ventilation. Pretty simple grab a manufacturer chart for me.

But anyhow if you were to investigate the power/utility companys recommendations and research, they pretty much recommend a continuos full width ridge vent and continious width eave inlets saying that is `best as far as naturally venting heat for the attic space.

A lot of that recomendation is based on direction of wind .. which can come from any directing and still completely evacuate the whole of the attic space. They would tell you eyebrow vents with wind blowing this way or that will at times be venting directly between themselfs and sometimes only within each vent as at some wind angles the opening will act as both intake and exhaust in the same opening doing very little and will not, except for during ideal wind directions, be actually sweeping the attic completely.

FHA and other governing bodies have specs and ratios to meet dependant on installed vapor and radiant bariers -- eave intake usually where most spec problems focus on.

I'm no expert,, just tossing in some I've been taught.

Have you reserched radiant barriers?
Check florida solar energy's site,
Also read building a home tech shield post.
Amazing the benefit of this install.

Well I took all yer comments ter heart. Altho I wanted a completely passive system in the end I installed a gable end fan and opend up tuther end etc. and removed the remaining w/Bird.

It became apparent that the vents true or effective xost areas were so small it would take too many vents to be realistic. The prevailing wind is from the South and the vents would face SW -- not so fine; they may or may not have worked.

I got a fan at Home Despot because it was the only one that would fit my existing geometry w/o a tremendous rebuild. Naturally it makes more noise than I personally like but it is tolerable. Presumably a larger diameter slower turning fan would move more air more quietly. (I usually put larger fans with reostats in my computer to drop the fan noise).

A fan will pull strongest from the nearest soffet, of course, so you have to play games and block off those nearest the fan to even out the flow. I use a hanging thread as a guage. Not really rigorous testing is it? but it doesn't have to be exact. Only a nut like me would go after it anyway. To that end today I blocked off with cardboard 12 soffet vents closest to the fan leaving 3 open 3 X 22 soffets, a 12 X 15 hole in one gable and another louvered hole 8 X 28 in tuther gable.

Well w/o the fan the attic temp was 142ºF at the mid point about 12 inches below the ridge. With the fan and all soffets open the temp was 130ºF. With the blocked-off soffets the temp was 125ºF. Ambient temp was 95º to 100ºF in all cases. We are having a cool summer. With blocked soffets there was noticeable displacement of the flow measuring device (thread) located near the thermometer.

The fan was 1500 CFM and they wanted at least 740 square inch inlet which is 5.1 square feet which would give an inlet velocity of 300 FPM which is 5 FPS which is not too damn very much. A fan of 3000 CFM would only give inlet velocity of 10 FPS. Again not to damn very much. My plan area is 1400 square feet and the attic volume about 1800 cubed feet so this fan should change the air in about 72 seconds. They are fearful of having too small an inlet. Well you are not going to get appreciable heat transfer out of the attic without velocity. This means a bigger CFM fan. My conclusion is that the fan CFM should be at least twice the volume of the attic. With vents open at the other end the inlet velocity and thru velocity will be higher therefore providing more cooling.

I can't honestly say there was a difference in the house probly cause there is lots of insulation. I do feel better knowing the attic is cooler. I spect that those claiming to feel a difference in the house probly don't have enough insulation.

We just replaced a shake roof with an ELK comp 40 year low impact - to get the insurance deduction. I had all penetrations removed except for 40 foot of Lomanco ridge vent. 40 foot provides 440 square inches of ventilation which is about 30% greater than the 1/300 requirement.
This gives the required "hot air rises" ventilation and works fine....

What helped greatly is that I also used OSB that had an aluminum radiant barrier attached to the underside. My attic has never been cooler ever.

Roof vents offer awesome value for money.