Ocean Dynamics. Equation of motion a=σf/ρ 29/08/11. What forces might cause a parcel of water to accelerate?

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1 Phsical oceanograph, MSCI 300 Oceanographic Processes, MSCI 5004 Dr. Ale Sen Gupta Ocean Dnamics Newton s Laws of Motion An object will continue to move in a straight line and at a constant speed unless acted on b a NET force The change in the velocit (speed and/or direction) of an object (e.g. a bit of water) is proportional to the force and inversel proportional to the mass of the object. a = ΣF but m = ρv m so, a = ΣF ρv Oceanographers will consider Acceleration a = So, = ΣF ρ = ΣF ρ for per unit volume. dw = ΣFz ρ u,: west to east v,: south to north w,z: up to down Equation of motion a=σf/ρ What forces might cause a parcel of water to accelerate? Movement of the ocean (a) = (/ρ)( wind - friction + rotation + tides gravit+ buoanc + pressure differences.) A sum of forces. = ( Fg + FC + FP + Ff ρ +...) Newton s Laws of Motion Vertical direction: P=F/A F top F bottom Weight (mg) z P P+ΔP What are the forces acting in the up-down direction? The boes weight is acting downwards (mg) The pressure at the top of the bo is also tring to force the bo downwards. But the pressure at the bottom of the bo is tring to force it upwards (the difference in the pressure forces is just the buoanc force discussed for Archimedes) Acceleration = /ρ(sum of forces) = /ρ(weight + F top - F bottom ) dw = g ρ dz Bouanc e to difference in pressures

2 Newton s Laws of Motion Vertical direction dw = ρ dz + g dw = ρ dz + g In general over the ocean vertical acceleration is much smaller than g. This means that in the vertical equation g and must be of similar magnitude ρ dz Vertical acceleration But we can normall make this even simpler. We can us SCALE ANALYSIS to see the size of the terms. e.g. Suppose A = B + C If we know that A= and B=0, then to a good approimation we could simplif this equation to B C Buoanc or pressure force Weight dz = ρg This is called the hdrostatic equation We can integrate this equation since densit and g are essentiall constant. p dz h dz = 0 0 ρgdz Or simpl p = hρg Which ou are hopefull familiar with alread! Newton s Laws of Motion Horizontal direction Hdrostatic equation tells us that pressure = weight of water above ou (in this case higher on the right) Acceleration = /ρ(sum of forces) = /ρ(f left - F right ) Barotropic and Baroclinic Motion Remember, p = ρgz, and = ρ p P P+ ΔP = ρ 0 5cm 5cm F right F left Or doing the same in the -direction: = ρ 0 d cm 2cm ρ 0 =027 2cm 2cm 2cm 2cm 2cm ρ =026 < ρ2=028 kgm -3 2

3 Barotropic and Baroclinic Motion Remember, p = ρgz, Barotropic Velocit is constant with depth and = ρ p Baroclinic velocit changes with depth Barotropic and Baroclinic currents Barotropic Currents move the whole water column. Can be inced b horizontal changes in elevation, which proce a pressure gradient throughout the whole water column. Baroclinic Currents var over depth. This can be inced b a horizontal densit gradient. This effect is known as the thermal wind balance (more later). BAROTROPIC BAROCLINIC ρ 0 ρ < ρ 2 Motion e to surface slopes Motion e to densit differences 0 Barotropic and Baroclinic Motion Remember, p = ρgz, and Mied situation ρ < ρ 2 = ρ p Motion e to densit differences Barotropic Ocean For a constant densit ocean, we can write the pressure gradient in an easier wa. Remember, p = ρgz, and = ρ d P =h ρg h =d+η So we are left with η η 2 Δ P 2 =h 2 ρg h =d+η η = g p p Δp p2 p = = Δ Δ h2ρg h ρg = Δ ( d + η2) ρg ( d + η) ρg = Δ ( η2 η) = ρg Δ p Δη = ρg Δ 3

4 29/08/ The Coriolis Force a= Coriolis hand out ΣF m p = + Coriolis force +... ρ The Coriolis Force Coriolis Force - Summar The Acceleration e to the Coriolis force is fv ( direction) and -fu ( direction) i.e the Coriolis Force the velocit It onl acts if water/air is moving (i.e a secondar force) Acts at right-angles to the direction of motion causes water/air to move to the right in the northern hemisphere The strength of the Coriolis force varies with latitude It is proportional to the Coriolis parameter f=2ωsin (Φ), where Φ is latitude And Ω is the angular velocit (in radians per second) It is maimum at the poles, zero at the equator and changes sign from NH to SH causes water/air to move to the left in the southern hemisphere E.g. Foucault s Penlum 4

5 The Equations of Motion d! u = " #! F Horizontal Equations: Acceleration = Pressure Gradient Force + Coriolis = " # + fv = " # d " fu = " #F = " #F Or, for a Barotropic Ocean: dw d# = "g + fv d# = "g d " fu = " # + fv = " #F z = " # d " fu If the onl force acting on a water parcel is Coriolis, the Navier Stokes equations can be simplified to: What does this mean? = fv = " fu u > 0 u < 0 v > 0 v < 0 Vertical Equation: Pressure Gradient force = Gravitational Force dz = ρg If the onl force acting on a water parcel is the Coriolis force, the Navier Stokes equations can be simplified to: = fv = " fu Northern Hemisphere: f > 0 If the onl force acting on a water parcel is the Coriolis force, the Navier Stokes equations can be simplified to: = fv = " fu Northern Hemisphere: f > 0 What does this mean? u > 0 u < 0 What does this mean? u > 0 u < 0 v > 0 v < 0 v > 0 v < 0 5

6 If the onl force acting on a water parcel is the Coriolis force, the Navier Stokes equations can be simplified to: What does this mean? = fv = " fu Southern Hemisphere: f < 0 u > 0 u < 0 Scaling arguments: = " # + fv = " # d " fu = ρg dz These are the equations ou need when there are both pressure and Coriolis forces in pla. But this is not alwas going to be the case. E.g. If there are no surface slopes or horizontal densit differences then there will be no pressure force (i.e. left with /=fv etc.). If we are on the equator there will be no Coriolis force v > 0 v < 0 Scaling arguments: Consider the situation where there are no pressure forces Acceleration = Pressure Gradient Force + Coriolis = " # + fv = " # d " fu If pressure gradients are small: = fv = " fu Inertia currents Acceleration = Pressure Gradient Force + Coriolis = " # + fv = " # d " fu If pressure gradients are small: = fv = " fu Inertia currents the water flows around in a circle with frequenc f. T=2π/f T(Sdne) = 2 hours 27 minutes 6

7 Scaling arguments: What forces are important in a bath tub? What kind of speeds will the water get up to? What kind of accelerations? What surface slopes? Need to look at the relative sizes of the terms in the equation. But first, choose which version of the equations are most appropriate. = " # + fv = " # d " fu d# = "g + fv d# = "g d " fu In a bathtub, densit is prett much constant, so conditions will be BARATROPIC Scaling arguments: What forces are important in a bath tub? What kind of speeds will the water get up to? What kind of accelerations? What surface slopes? d# = "g + fv d# = "g d " fu Size of the pressure force: dη = 0.m /m = 0. dη g = 0(0.) = ms Size of the Coriolis force: fu = = 70 ms So Coriolis<<Pressure, so we can neglect rotation effects dη g 2 Acceleration in the bathtub is driven b pressure differences (e to changes in surface slopes) Scaling arguments:. If there are no surface slopes or horizontal densit differences then there will be no pressure gradient force (i.e. left with /=fv, /=-fu, inertia currents): = fv = " fu 2. If ou are sitting in our bathtub, ou are in a barotropic environment AND the Coriolis force can be neglected: d# = "g d# = "g d But the ocean is not a bathtub. We will conct a scaling analsis on our equations of motion... to find further simplifications for motions with a period greater than ~0 das Scaling Analsis: = " # + fv T~0 das = s ~ 0 6 s = " # d " fu u,v ~ U ~ cms - - ms - f~ 0-4 s - 7

8 But the ocean is not a bathtub. But the ocean is not a bathtub. We will conct a scaling analsis on our equations of motion... to find further simplifications for motions with a period greater than ~0 das Scaling Analsis: = " # + fv T~0 das = s ~ 0 6 s = " # d " fu u,v ~ U ~ cms - - ms - f~ 0-4 s - We will conct a scaling analsis on our equations of motion... to find further simplifications for motions with a period greater than ~0 das = " Scaling Analsis: # + fv T~0 das = s ~ 0 6 s = " # d " fu u,v ~ U ~ cms - - ms - f~ 0-4 s - Acceleration << Coriolis Geostrophic Balance Acceleration is much smaller than Coriolis and Pressure Gradient Force (this is true almost everwhere in the ocean) The ocean is in Geostrophic Balance (= balance between Pressure Gradient and Coriolis Forces) Pressure forced motion Geostrophic Transport Ingredients: () Pressure force acts from high pressure to low pressure (2) Coriolis alwas tries to push a moving object to the left (SH) or right (NH) Ocean is in Stead State (no acceleration) / is negligible = " # + fv = " # d " fu " = fv " d = # fu L Imagine that somehow water has been piled up in some area e.g. b the ac7on of winds. What happens when the wind stops? H 8

9 Pressure forced motion Geostrophic Current Ingredients: () Pressure force acts from high pressure to low pressure (2) Coriolis alwas tries to push a moving object to the left (SH) or right (NH) Pressure forced motion Geostrophic Current Ingredients: () Pressure force acts from high pressure to low pressure (2) Coriolis alwas tries to push a moving object to the left (SH) or right (NH) SH Eample (looking down on the ocean) SH Eample (looking down on the ocean) L Pressure Force H L Water flow H Pressure forced motion Geostrophic Current Ingredients: () Pressure force acts from high pressure to low pressure (2) Coriolis alwas tries to push a moving object to the left (SH) or right (NH) Pressure forced motion Geostrophic Current Ingredients: () Pressure force acts from high pressure to low pressure (2) Coriolis alwas tries to push a moving object to the left (SH) or right (NH) SH Eample (looking down on the ocean) SH Eample (looking down on the ocean) L H L H Coriolis Force Coriolis Force 9

10 Pressure forced motion Geostrophic Current Ingredients: () Pressure force acts from high pressure to low pressure (2) Coriolis alwas tries to push a moving object to the left (SH) or right (NH) Pressure forced motion Geostrophic Current Ingredients: () Pressure force acts from high pressure to low pressure (2) Coriolis alwas tries to push a moving object to the left (SH) or right (NH) SH Eample (looking down on the ocean) SH Eample (looking down on the ocean) L H L H Pressure forced motion Geostrophic Current Ingredients: () Pressure force acts from high pressure to low pressure (2) Coriolis alwas tries to push a moving object to the left (SH) or right (NH) SH Eample (looking down on the ocean) L Pressure Force Pressure Force = Coriolis Force So, no net force on the water So keeps on going H Coriolis Force What does the geostrophic balance mean phsicall? Geostrophic Balance: How is it set up? Suppose we have a difference in sea-level height. Water will want to move from the region of high pressure towards the region of low pressure. Etends down to the deep ocean Water flow 40 0

11 Geostrophic Balance: How is it set up? (cont) As the water starts to move, the Coriolis effect (rotation) deflects the water to the right (NH) or left (SH). The water keeps getting deflected until the force e to the pressure difference balances the Coriolis force. This balance is called a geostrophic balance and the resulting current is referred to as a geostrophic current. 4 Geostrophic Currents Geostrophic Edd (Northern Hemisphere) High pressure Low pressure horizontal pressure gradient force 42 Which direction is the Geostrophic wind? (f <0 SH) Concept Problem (NH): In the NH: Which wa does the current flow if sea level height is increasing towards the South? West? North? PG CF V dη fv = g dη fu = g d

12 Concept Problem (NH): In the NH: Which wa does the current flow if sea level height is increasing towards the South? You can use the equations, or just think about the forces Geostroph Problem 2: Which direction does the water flow around this pressure feature if it is in the Northern Hemisphere? NH: f>0 dη/d<0 u=(-)(+)(+)(-)>0 East! dη fv = g dη fu = g d C P Geostroph Problem 2: Which direction does the water flow around this pressure feature if it is in the Northern Hemisphere? Geostroph Problem 3: Which direction does the water flow around this pressure feature if it is in the Southern Hemisphere? 2

13 Geostroph Problem 3: Which direction does the water flow around this pressure feature if it is in the Southern Hemisphere? Northern Hemisphere Anticclone Cclone clockwise counterclockwise Southern Hemisphere Anticclonic circulation: ² ALWAYS around a high pressure sstem. Geostroph Problems: A certain ocean current has a height change of. m (increasing to the east) over its wih of 00 km at 45 N. How fast is the current flowing? ² clockwise in the Northern Hemisphere ² counter-clockwise in the Southern Hemisphere dη Δη fv = g g Δ f=0-4 s - g=0ms -2 Δη=.m Δ=00000m Cclonic circulation: ² ALWAYS around a low pressure sstem. ² counter-clockwise in the Northern Hemisphere ² clockwise in the Southern Hemisphere V= m/s Summar: Geostroph is the balance between pressure forces and Coriolis. In most of the open ocean (awa from boundaries) motion will become geostrophic after a few das. Geostroph doesn t work over short periods of time or small distances (other forces become dominant). Geostroph also fails in regions where friction becomes important 52 3

14 Recap = " # o + fv = " # o d " fu dz = #g } Acceleration = Pressure + Coriolis Force Weight of water is balanced b its buoanc (or vertical pressure force) Recap = " # o + fv =0 = " # o d " fu =0 dz = #g } Acceleration = Pressure + Coriolis Force Weight of water is balanced b its buoanc (or vertical pressure force) After a period of time flow becomes stead, so, acceleration becomes zero. This occurs when the pressure force balances the Coriolis force After a period of time flow becomes stead, so, acceleration becomes zero. This occurs when the pressure force balances the Coriolis force = ( Fg + FC + FP + Ff ρ +...) The last force to consider is friction. This is onl important at continental boundaries, at the bottom of the ocean, and at the surface (e to wind) What will the friction term look like? We know that friction alwas tries to retard motion. So far we have neglected friction. Effects of Friction A simple model for the frictional force at the sea floor in the and direction is: ru rv and h h h is the depth and r is the dissipation constant Raleigh frictional dissipation, r is a coefficient (r ~ 0-7 s - ) / =? Hence the equations of motion become : d# = "g + fv " ru h d# = "g d " fu " rv h 56 4

15 To eamine the effects of friction, consider the simple balance: ru = h A solution is: So that at t=0, u=u o u u e rt / h = h/r represents the time o it takes for the speed to drop to about /3 of its initial value U i.e no pressure forces and no coriolis. Motion is just decelerated because of friction Thermal Wind Balance So far we have assumed that densit ρ is constant (barotropic) Small horizontal changes in ρ can result in large vertical changes in current/ wind e.g. near fronts and eddies Baroclinic velocit changes with depth Velocit decreases with time because of friction Known as the e-folding time scale of frictional spin down. U o /3U o U o e - h/r Time 57 ρ < ρ 2 Motion e to densit differences Thermal Wind Balance So far we have assumed that densit ρ is constant (barotropic) Small horizontal changes in ρ can result in large vertical changes in current/ wind e.g. near fronts and eddies Thermal Wind Balance So far we have assumed that densit ρ is constant (barotropic) Small horizontal changes in ρ can result in large vertical changes in current/ wind e.g. near fronts and eddies Starting with the geostrophic balance " = fv u d" << " " d = # fu dz dz We can differentiate the equations with regard to depth (z), then substitute for /dz v = "f u = # "f d dz = d # & dz $ "f ' dz = d # dz ) & $ "f d ' dz = "f dz = ) "f d # & dz $ ' d # & dz $ d ' dz = "f dz = ) "f d # & $ dz ' d # & d $ dz ' 5

16 Thermal Wind Balance So far we have assumed that densit ρ is constant (barotropic) Small horizontal changes in ρ can result in large vertical changes in current/ dz = d ( "f "g) wind e.g. near fronts and eddies Starting with the geostrophic balance dz = # "f u d" << " dz dz We can differentiate the equations with regard to depth (z), then substitute for /dz v = "f u = # "f d dz = d # & dz $ "f ' dz = d # dz ) & $ "f d ' dz = "f dz = ) "f d # & dz $ ' d # & dz $ d ' dz = "f dz = ) "f d d "g ( ) d # & $ dz ' d # & d $ dz ' Thermal Wind Balance So far we have assumed that densit ρ is constant (barotropic) Small horizontal changes in ρ can result in large vertical changes in current/ dz = d ( "f "g) dz = g d" "f wind e.g. near fronts and eddies Starting with the geostrophic balance dz = # "f dz dz We can differentiate the equations with regard to depth (z), then substitute for /dz v = "f u = # "f d u d" << " dz = d # & dz $ "f ' dz = d # dz ) & $ "f d ' dz = # g d" "f d dz = "f dz = ) "f d # & dz $ ' d # & dz $ d ' dz = "f dz = ) "f d d "g ( ) d # & $ dz ' d # & d $ dz ' For geostrophic conditions: Thermal Wind Balance The vertical structure of u and v is related to the horizontal densit gradients dz = g "f dz = # g "f d" d" d Thermal Wind Balance i.e. horizontal densit gradients in temperature (T) and salinit (S) can eplain the change in horizontal velocit with depth (vertical profile of horizontal velocit). Velocit is into the page 6

17 Thermal Wind Balance Thermal Wind Balance dz = g d" "f NH: f > 0 d" > 0 dz = g d" "f d" > 0 NH: f > 0, so, dz > 0 Velocit is into the page Velocit is into the page and increases with depth i.e. v gets more positive with increasing depth cold/saline core Potential Densit through an Edd near the Gulf Stream. How does the geostrophic current velocit in the edd change with depth? 7

18 dz = g d" "f dz = # g d" "f d dz = g "f d" # g "f $" $ dz = g d" "f dz = # g d" "f d dz = g "f d" # g "f $" $ dz = g d" "f dz = # g d" "f d Estimate the densit gradient at =40km: ρ changes from to kgm -3 over 25 km. dz = g "f d" # g "f $" $ dz = g d" "f dz = # g d" "f d Estimate the densit gradient at =40km: ρ changes from to kgm -3 over 25 km "# $027.2 = = $.6 %0 $5 " 25km dz = g % $.6 %0$5 #f = $.5 %0 $3 s $ 8

19 dz = g "f d" # g "f $" $ This means that at z = 500m, dz = ".5 #0"3 s " let s see how much v varies over 00m of depth at z=500m (Δz=00m, Δv=?): dz = g "f d" # g "f $" $ This means that at z = 500m, dz = ".5 #0"3 s " let s see how much v varies over 00m of depth at z=500m (Δz=00m, Δv=?): "v = #.5 $0 #3 s # $00m = #0.5ms # = #5cms # dz = g "f d" # g "f $" $ This means that at z = 500m, dz = ".5 #0"3 s " let s see how much v varies over 00m of depth at z=500m (Δz=00m, Δv=?): "v = #.5 $0 #3 s # $00m = #0.5ms # = #5cms # i.e the velocit shear ~ -5 cm/s per 00m depth increase. To work out the actual velocities, we need one more piece of information: at depth (sa 2000m) the velocit is 0. This is called the depth of no motion This means that v (velocit into page) is getting more negative as we get deeper

20 Going down in depth, the densit surfaces flatten out. We can assume a level of no motion where there is no longer a change in densit. Hence we can calculate the change in velocit up through the water column. Note that on the other side of the edd, the densit surfaces slope the other wa, so the circulation must also be in the opposite sense. Going down in depth, the densit surfaces flatten out. We can assume a level of no motion where there is no longer a change in densit. Hence we can calculate the change in velocit up through the water column. Note that on the other side of the edd, the densit surfaces slope the other wa, so the circulation must also be in the opposite sense. V=0 Going down in depth, the densit surfaces flatten out. We can assume a level of no motion where there is no longer a change in densit. Hence we can calculate the change in velocit up through the water column. Note that on the other side of the edd, the densit surfaces slope the other wa, so the circulation must also be in the opposite sense. Going down in depth, the densit surfaces flatten out. We can assume a level of no motion where there is no longer a change in densit. Hence we can calculate the change in velocit up through the water column. Note that on the other side of the edd, the densit surfaces slope the other wa, so the circulation must also be in the opposite sense. V=0+.5 V=0 V is positive into the page 00m V= V=0+.5 V=0 V is positive into the page 00m 00m 20

21 Going down in depth, the densit surfaces flatten out. We can assume a level of no motion where there is no longer a change in densit. Hence we can calculate the change in velocit up through the water column. Note that on the other side of the edd, the densit surfaces slope the other wa, so the circulation must also be in the opposite sense. A phsical eplanation NH f>0 V= V= V=0+.5 V=0 V is positive into the page 00m 00m 00m v = ρf A phsical eplanation NH f>0 NH f>0 Step : denser in the middle so the surface will be depressed v = ρf v = ρf 2

22 NH f>0 NH f>0 P Step 2: Start b figuring out the pressure force e to the surface slope: Pressure increases moving awa from centre Opposite on the other side of the edd Step 3: Forget the surface pressure part. What happens at greater depth? Densit is higher at the centre than further to the right. So the pressure force, just e to densit would be in the positive -direction (on the right hand side of the gre). The densit gradient causes a pressure force that increases with depth Remember the fishtank eperiment Remember the fishtank eperiment Opposite happens on the left side of the gre ρ < ρ 2 NH f>0 NH f>0 Step 3: Keep repeating this down the water column, until there is no densit gradient (i.e. at the bottom of the cold core edd) But a smaller pressure force means a smaller geostropic velocit (going into the page) with depth Step 3: So if we add up the forces e to the surface slope and e to the densit gradient we get a pressure force that decreases with depth v = ρf Remember the fishtank eperiment 22

23 NH f>0 NH f>0 Step 3: Keep repeating this down the water column, until there is no densit gradient (i.e. at the bottom of the cold core edd) But a smaller pressure force means a smaller geostropic velocit (going into the page) with depth C P P C Finall we need to use the geostrophic relationship: If we have a pressure gradient in the direction (as we do), it will create a geostrophic velocit in the direction, proportional in strength to the pressure gradient. v = ρf v = ρf Eample: Derive the rotation of these cold and warm water eddies, in the SH, using the thermal wind balance. Eample: Derive the rotation of these cold and warm water eddies, in the SH, using the thermal wind balance. 9 23

24 Thermal Wind Balance For geostrophic flow (i.e. pressure is balanced b Coriolis): Geostrophic flow in the presence of horizontal densit gradients Horizontal densit gradients (T,S) can eplain vertical velocit changes To know absolute velocit we need etra information (i.e. we need to know absolute velocit at some depth) dz = g "f dz = # g "f d" d" d Given that u & v = 0 at 2000m (depth of no motion) estimate the surface current. - Can figure out surface velocit from surface heights. - Often we assume that at a certain depth (e.g. 2000m) velocities are zero this is called the depth of no motion. - Once we know the velocit at the surface or the depth of no motion we can calculate velocit at all other depths using the thermal wind equation. Summar: Ocean Dnamics Most of the motion in the ocean can be understood in terms of Newton s Law that the acceleration of a parcel of water (how fast its velocit changes with time /) is related to the sum of forces acting on that parcel of water. We can split the forces, velocities and accelerations into south-north (,v), west-east (,u) and up-down (z,w) components. In the vertical direction the acceleration is related to the difference between the water weight and the bouanc (or pressure) force. When there is a vertical densit gradient this leads to oscillations (Brunt Väisälä frequenc N). The densit gradient tries to inhibit vertical motion (and miing). These vertical accelerations are generall ver weak, so we get the hdrostatic equation. If the hdrostatic equation is integrated over depth, it just sas that the pressure at a point just equals the weight of water above that point. = ρg dz Acceleration in the horizontal can be driven b a number of different forces: () The pressure gradient force. This eists whenever there is a surface slope and/ or a horizontal densit gradient. (2) Coriolis (because we live on a rotating planet). It is ver weak, so we onl feel its effect over long times (> few das) and large distances (> 0s of km). Coriolis onl affects moving fluids, deflecting to the right in the NH and to the left in SH. (3) Friction. Also onl acts on moving water. Alwas acts to slow down motion. Important at the boundaries of the ocean. Summar: Ocean Dnamics = " + fv " ru # = " " fu " rv # d = ρg dz Or for a constant densit (barotropic ocean): d# = "g + fv " ru d# = "g " fu " rv d Over much of the ocean, the flow is stead (i.e. /=/=0) and friction is negligible, so we are left with the geostrophic balance i.e. pressure gradient forces balance coriolis. The current moves at right angles to the pressure gradient. P C " = fv, " d = # fu When there is a horizontal densit gradient the velocit changes with depth. This can be calculated using the thermal wind equations dz = g d" "f, dz = # g d" "f d 24