Process Studies – Part I

general concepts of the basin-scale, buoyancy-drivencirculation, including:

forcing and ocean response simple two-dimensional (depth-latitude) modelsenergetics of thermohaline flowsimportance of diapycnal mixingbasic concepts of the abyssal circulation

(two active tracers and multiple equilibria)

thermohaline “Ocean Conveyor” circulation

Useful as very general schematic to convey that:- the ocean circulation is of global scale- the ocean circulation is three dimensional- the ocean transports heat (and other properties)

between basins and across latitude circles

Much of what we will talk about in this school will providedynamically based descriptions of the key elements ofthis cartoon.

A more detailed schematic of the buoyancy-forcedoceanic thermohaline circulation

Key elements:1. Transformation at high latitudes (deep-convection)2. Sinking due to mixing near boundaries

overflows and entrainmentupper ocean

3. Upwelling balanced byinterior diapycnal mixingSouthern Ocean Ekman suction

4. Upper ocean return flow

(Huang, 2010)latitude

depth

Temperature/Salinity census of cold waters in the world ocean(height represents volume )

The coldest waters are dominated by a few water types

NADW and AABW may be traced to surface properties atformation sites

Mixing between sources is evident

(Worthington, 1981)AABW

NADW

Sea Surface Temperature

Generally warm at low latitudes, cold at high latitudesSharp gradients coincident with western boundary currentsDeep convection is isolated to a few locations :

- Labrador Sea, also Irminger Sea- Nordic Seas- Mediterranean Sea, Red Sea- Weddell Sea

The ocean is (generally) warmed at low latitudesand cooled at high latitudes

Enhanced heat loss over the warm western boundary currents

(Talley, 2000)

Annual mean net surface heat flux into the ocean

Blue: heat loss

Pink: heat gain

The net heat flux through the surface is balanced bylateral advection in the ocean

The ocean advects heat from low to high latitudes inmost basins (total approximately 2 x 1015 W)The ocean gains heat near the equator, loses heatpoleward of 20 S and N

The exception is the Atlantic Ocean, where the heat flux istowards the north in both the South and North Atlantic

Schematic meridional section of temperaturein the North Atlantic

Warm thermocline waters confined near surface

Surface waters move northward, are cooled, and sinkto form North Atlantic Deep Water

Densest waters originate in Southern Ocean, AABW

What can we learn about the buoyancy-forced circulation from simple models?

Let’s start with a very simple two-dimensional model(following Rossby 1998)

b is the buoyancy of the fluidHeat at low latitudes and cool at high latitudesInsulate the lateral boundaries and bottomAssume the flow is two-dimensional (latitude/depth)

latitude

depth

The governing equations :

0

2

2

=⋅∇

∇=

∇++−∇=×+

v

bDtDb

vAkbvfDt

vD

κ

φMomentum

Buoyancy

continuity

If we consider the zonally averaged flow, we can define astreamfunction as

∂∂

+∂∂

=∇=

∂∂

=∂∂

−=

2

2

2

22

zy

yw

zv

ψψψζ

ψψ

The vorticity is then

Take the curl of the zonally averaged momentum equations:

( )

( )

( )yb

za

zb

yabaJ

bbJtb

AybJ

t

∂∂

∂∂

−∂∂

∂∂

=

∇=+∂∂

∇+∂∂

=∇+∂

∂∇

,

,

,

2

422

κψ

ψψψψVorticity equation

Buoyancy equation

Jacobian operator

**

***

tLHtHzz

LyyHLbbb

ψ

ψκψ

==

==∆=

Now nondimensionalize the variables with :

∆b is buoyancy difference across surfaceH is vertical scale of the basinL is horizontal scale of the basin

Streamfunction is scaled in anticipation of advective/diffusivebalance in buoyancy equation

Time is scaled with an advective time scale

Nondimensional equations :

( )

( ) bbJtb

ybRaJ

t2

4522

,

,

∇=+∂∂

∇+∂∂

=∇+∂

∂∇

ψ

ψσασψψψ

With three nondimensional numbers:

Raleigh number

Prandtl number

Aspect ratioLH

AAbLRa

=

=

∆=

α

κσ

κ

3Ra small, diffusion is important

Solutions for three values of Raleigh number

streamfunction

buoyancy

Ra = 104 106 108

1. The horizontal circulation is basin-scale2. downwelling is confined to a narrow region

upwelling is found over a broad region3. most of domain filled with dense water, the

stratification is trapped closer to the surface

How is this circulation maintained ?

Derive a kinetic energy equation

( ) vvAwbbvt

v

22

21

∇⋅++⋅−∇=∂

∂

Integrate over the domain, assume no stress at boundaries

vvA

wbvt

2

2

21

∇⋅−=

−=∂∂

ε

ε

ε is the total dissipation of kinetic energy, positive definite

Now consider the potential energy equation

wbQzDt

Dbz+=

wbQzbzdtd

+=

ε−−=− Qzbzvdtd

2

21

Integrate over the domain, with insulated boundaries, to get

Subtract from KE equation to get total energy equation

Change in total energy is given by net buoyancy sourcesminus kinetic energy dissipation

Q is heat sourceor sink

In a statistically steady state :

0<−= εQz

The implications are significant

RHS is negative definiteIn order to balance kinetic energy dissipation, the heatingmust be negatively correlated with height

Heating must take place at a lower level than cooling

This is true of the atmosphere (heating at surface, coolingin mid-atmosphere

This is not true of the ocean, heating and cooling bothtake place at the surface (geothermal is negligible)

In order for their to be buoyancy-forced turbulent motion inthe ocean, the heating must penetrate below the surface.

Vertical diffusion of heat down from the surface can actlike an interior source of heat, allowing for turbulent motion

Thus vertical diffusion is required to drive a purelybuoyancy-forced circulation in the ocean.(Mechanical forcing, such as by winds or tides, breaks thisconstraint.)

Mixing warm water downward increases the potentialenergy of the water column, which is ultimatelyconverted to kinetic energy and dissipated.

How much energy is required, and where does it come from?

We know there is about 30 Sv of overturning in the oceanOver the whole ocean, this gives a mean vertical velocityof about w=10-7 m/sThe observed change in density, top to bottom, is ∆σ = 1 Kg/m3

The downward diffusion must balance vertical advection, so

The total energy is obtained by multiplying by the area ofthe abyssal ocean, 3x1014 m2 to get

1x1012 W = 1 TW

A very rough estimate of the energy required can be obtainedby considering how much warm water is diffused downward

∫−

−

−− ×≈∆2000

5000

2310 /103 mWdzwCp σαρ

The sources and sinks of energy in the ocean arenumerous and the uncertainties are very large, but …

(Wunsch andFerrari, 2004)

Main points:1. Largest sources of energy are winds and tides2. Heating/cooling, E-P, and geothermal are very small3. Enormous energy storage in general circulation4. Energy required to maintain overturning circulation is

much smaller than energy input to the oceanenergy transported northward by ocean circulation

So how does the thermohaline circulation depend onthe strength of diapycnal mixing ?

Lets consider some scaling analysis:

Assume basin of horizontal scale LMeridional temperature difference of ∆TDepth of the thermocline δTCoriolis parameter f

Thermal wind gives a horizontal velocity scale

fLTgU Tδα∆

=

g is gravitational accelerationα is the thermal expansion coefficient

(1)

Continuity gives the scale of the vertical velocity

LUW Tδ

=

zzz TwT κ=

It is assumed that the upwelling in the interior is balancedby downward diffusion of heat

horizontal advection of density is neglected(this is a good approximation if U/βL2 << 1)

This yields a different estimate of the vertical velocity

T

Wδκ

=

(2)

(3)

Equations (1), (2), and (3) can be combined to give

( ) 3/1

2

2

3/12

∆=

∆

=

LfTgU

TgfL

T

ακ

ακδ

The depth of the thermocline and the strength of thehorizontal circulation both depend on the strength ofdiapycnal mixing κ

Note also that they also depend on the equator-to-poletemperature change. Assume for now that they areindependent of each other (not necessarily the case).

Now we can get estimates for the overturning strength3/142

∆==Ψ

fTgLLU T

ακδ

And the meridional heat transport3/1442

00

∆=Ψ∆=Θ

fTgLCTC pp

ακρρ

Both of these climatically important quantities depend onthe mixing coefficient

These scaling estimates can be tested with a numerical model(Park and Bryan, 2000)

60 degree wide sector, equator to 60 degrees latitude

2o x 2o horizontal grid (both z-coordinate level and layer models)

The model is forced by a surface heat flux that is linearlyproportional to the air-sea temperature difference with timescale τT = 30 days

Tsstairsp TThCQ τρ /)( −=

The model is initialized at rest and run for several thousandyears. Long times are required to achieve thermal equilibrium

)1000(1010

4

62

yearsOOH=

== −κ

τ

A series of model calculations were carried out in whichκ was varied from .05x10-4 to 10x10-4 m2/s

Scaling law is reproduced well in the model

Note that this uses the diagnosed ∆T at the ocean surfaceinstead of the specified ∆Tair

Using ∆Tair does not compare as well because SST depends on κ

Note meridional heat transport >> mixing energy (1012 W)

The maximum of the overturning streamfunction,( diagnosed in density coordinates)

Compares reasonable well with scaling theory

Demonstrates that the overturning circulation is stronglydependent on the ability of mixing to balance upwellingof dense water

Thickness of the thermocline

Scaling law reproduced well in the model

Much weaker dependence on mixing

Note that the amount of energy used to mix is differentfor each of these cases.

If the energy available for mixing is drawn from wind andtides, we might expect that the energy is fixed and the mixing varies.

Nilsson et al. (2003) tested this dependence using a numerical model. They configured the model with 3types of mixing:

A: fixed κB: κ proportional to N-1

C: fixed energy for mixing

Calculations with a GCM suggest that the sensitivityof the THC to equator-to-pole temperature differencedepends on how mixing is treated

Maximum MOC diagnosed froma series of model calculationsin which ∆T was varied

The strength of the MOC gets weaker for larger ∆T formodels B and C! This is because the mixing is reducedfor increased stratification, and MOC is more sensitive tomixing strength than it is to ∆T

(Nilsson et al., 2003)

So far, we have concentrated on zonally averaged aspectsof the thermohaline circulation. Let’s consider a verysimple model of the horizontal circulation driven by sources and sinks of dense water in the abyss.

The underlying theory has its origins in the upper ocean.Consider the flow on a beta-plane

Goldsborough (1933) recognized the linear vorticity balance inthe ocean interior for a net precipitation P with ocean depth DThis is a freshwater version of the wind-driven Sverdrup balance.

yff β+= 0

DPfv 0=β

The horizontal circulation driven by P could be calculatedfrom the continuity equation. The zonal velocity could goto zero at the eastern and western boundaries only if

Sverdrup (1947) recognized that precipitation P could bereplaced by the Ekman pumping rate to consider the wind-driven circulation. Introduction of the westernboundary current (Stommel, 1948; Munk, 1950), withdissipative dynamics, allowed for closure of the circulationwith a net vorticity input and interior poleward flow.

DPfv =β

DPvu yx −=+

00

=∫X

dxP

We know that in the basin interior, the downward diffusionof heat is balanced, to leading order, by upwelling of the mean stratification. We can use similar dynamical balancesto infer the horizontal circulation driven by sources andsinks of mass to the abyssal ocean by replacingP with w in the interior (Stommel).

zzz TwT κ=

For simplicity, we will represent the abyssal ocean asa single layer of uniform density.

Not a terrible assumption, but adding stratification addscomplexity (baroclinic layering flows)

Various consequences and flow patterns can be demonstrated in a simple laboratory experiment (Stommel, Arons, and Faller, 1957)

Consider a pie-shaped container of fluid rotating at Ωwith width Θ0 and radius a

Because the tank is rotating, the free surface is not flat

grhD

2

22

0Ω

+=

We can think of this laboratory basin as a very idealizedrepresentation of the North Atlantic Ocean with a highlatitude source and upwelling in the abyssal interior

More detailed discussions on where/how waters sinknext week

Fluid is added to the basin at rate S. Because there is nosink of fluid, the basin slowly fills. The rate of change ofthe surface height (analogous to w) is

022Θ

=a

Sht

00

22ρρ

θ rgDur

gDv −=Ω−=Ω−

The flow in the interior is in geostrophic balance

The continuity equation is written as :

0)()(1=++ rt vDuD

rh θ

Continuity, continued…

( ) ( ) 0=+++ rrt vDvDr

uDh θ

0=+ rt vDhBecause the flow is geostrophic, terms 2 and 3 cancel, so

Making use of the upwelling relation to the source strength,

ΘΩ−= 22

2a

Sr

gv

For S > 0, v < 0, or towards the apex of the basin, always!

This is independent of where the source is located

Consider the mass balance in the region r < r0There is the source S (inflow)There is the interior flow towards the apex (inflow)There is the rise in height of the fluid (outflow)

source + interior - rise

2

20

0000 )()(arSrrDrvS −Θ−

This will, in general, not be equal to zero. How is themass balanced ?

Ω+= 22

021a

ghSTw

- Tw > S, some of the western boundary current isrecirculated from the interior

- Higher order boundary layers are required for both thewestern boundary current and the radial boundary current(as for upper ocean wind-driven flow)

Introduce a western boundary current (as in wind-driven theory)

Now consider a case in which the source is at radius a

Ω

= 2202

aghSTw

The western boundary current transport is everywheretowards the source! All western boundary currentwater is recirculated from the interior

What if we balance the source with a sink ?

both on the western bdy

1. v=0 in the interior because w=0 there2. boundary layer dynamics can not support a viscous

eastern boundary current3. the result is a unidirectional flow from source to sink

Put the source and sink on the eastern boundary

both on the western bdy both on the eastern bdy

1. v=0 in the interior because w=02. boundary layer dynamics can not support a viscous

eastern boundary current3. The only way to close the mass balance is by

having zonal jets connect the eastern boundarysource/sink with the viscous western boundary, nomatter how far away the western boundary is

So, how can this simple model be applied to the real ocean?

1. Assume uniform upwelling in the basin interior2. Use similar dynamical assumptions on a sphere3. Assume two sinking regions: North Atlantic

Weddell Sea

Require assumption of weak bottom topography(no closed f/h contours)

This example uses a flat bottom

So, how can this simple model be applied to the real ocean?

1. Assume uniform upwelling in the basin interior2. Use similar dynamical assumptions on a sphere3. Assume two sinking regions: North Atlantic

Weddell Sea

(Stommel, 1957)

Such interior poleward flows have not been observedDistribution and direction of DWBCs are consistent with observations, major success of the theory

Summary of keys points :

1. Buoyancy-forcing of the ocean occurs at the surface2. Downwelling is confined to narrow regions near boundaries3. Upwelling is more broadly distributed

(we have not considered wind-forcing in Southern Ocean)4. Maintenance of the buoyancy-driven flow requires

diapycnal mixing5. Energy source for mixing is most likely wind and tides6. Two dynamically active tracers (T and S) can give rise

to complex solutions (both steady and time-dependent)7. Strong horizontal circulations can be forced by weak

vertical motions, patterns are sometimes non-intuitive

Now for something completely different…Let’s consider a simple model with 2 active tracers, T and S

(Stommel, 1961)

1. Two reservoirs of uniform temperature and salinity( T1 and S1 ) and ( T2 and S2 )

2. Connected by two pipes, overflow and deep return flowwith circulation strength Ψ

3. The system is forced by relaxation of temperature andsalinity in each reservoir towards + or – T* and S*

What does this strange system represent in the real ocean ?

We interpret the cold reservoir as representing theentire high latitude ocean and the warm reservoiras representing the entire low latitude ocean.

The capillary tube is the analog of the DWBC,the overflow tube is the analog of the shallow northward flow

The reservoirs T* and S* represent the atmosphere,cold and fresh (precipitation dominates) at high latitudes,warm and salty (evaporation dominates) at low latitudes.

T* and S* have the opposite affect on the ocean density

Note that we assume each box is well mixed, ignore vertical stratification.

This is a crude parameterization of atmospheric forcing,a cold ocean is heated, a warm ocean is cooled

Note: A similar process is assumed for salinity but using a different constant, d

Assume that the strength of the flow between the tworeservoirs is proportional to the pressure gradient, whichbecause the flow is hydrostatic, is also proportional to thedensity difference between the two boxes (key assumption!)

)( 21 ρρ −=Ψ A

If the temperature in the reservoir is different from T*, it isrestored towards T* with rate c

)( 1* TTcQ −=

We can form conservation equations for each variable ineach reservoir

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )122*2

211*1

122*2

211*1

22

22

SSSSdt

SSSSSdtS

TTTTct

TTTTTctT

−Ψ−−−=∂

∂−Ψ−−=

∂∂

−Ψ−−−=∂

∂−Ψ−−=

∂∂

*21

*21

22 SSSS

TTTT −

=−

=

Note that the absolute T and S do not matter, only their difference

We can define new non-dimensional variables :

)1(0 ST ST ββρρ +−=Assume a linear equation of state:

Nondimensionalize time with time scale 1/c, we have twoconservation equations for T and S

( )

( )

)(

1

1

ST

SStS

TTtT

µγ

δ

−−=Φ

Φ−−=∂∂

Φ−−=∂∂

where Φ=4Ψ/c and there are three nondimensional numbers:

*

**08

TS

cd

cAT

T

ST

ββµδβργ ===

What do these non-dimensional numbers mean?

*

**08

TS

cd

cAT

T

ST

ββµδβργ ===

γ is the ratio of a relaxation time scale to an advective time scalefor γ < 1 the atmospheric forcing dominatesfor γ > 1 ocean advection dominates

δ is the ratio of relaxation time scales for T and Sin reality, salinity is forced by E-P, independent of Swhile SST is “relaxed” towards an atmospheric temperaturewe will represent these different physics by taking δ << 1

µ is the ratio of the relative strengths of T and S forcing on density

Let’s look at some steady state solutions. The threeequations can then be combined for a single variable Φ

)(/11

1Φ=

Φ++

Φ+−

=Φ gδ

µγ

For parameters:γ = 5 (advection dominates surface forcing)δ=1/6 (temperature relaxation is faster than salt relaxation)

µ=0.5 atmospheric change in T larger than change in S1.0 atmospheric change in T and S are the same1.5 change in S is larger than change in T

First consider when temperature dominates µ=0.5Solutions are found when Φ = g(Φ)

One solution at Φ=−1.3For negative Φ, the circulation is from warm to coldin the upper ocean (poleward), with a cold, deep returnThis is analogous to the present day circulation in theNorth Atlantic Ocean.

The temperature component dominates the densityand controls the direction of flow

Temperature and salinity change equal, µ=1.0 (red line)

Still only one solution at Φ=−1.3, but there is anindication that more solutions will emerge as salinity forcing becomes stronger

Salinity change is greater than temperature change, µ=1.5(blue line)

Now there are three possible solutions:Φ=−1.2 temperature dominates, strong poleward surface flow

Φ=−0.1 temperature > salinity, weak poleward surface flow(turns out this solution is unstable)

Φ=+0.1 salinity dominates, weak equatorward surface flowhigh salinity at low latitudes, low salinity at high lats

The most interesting aspect of this model is that issupports more than one equilibrium solution for thesame forcing parameters.

If salinity contrasts are sufficiently large, salinity can dominate temperature and control the direction of flow.

This simple model suggests some interesting, and complexbehavior when there are two active tracers. This studyhas prompted much research, and similar behavior hasbeen found in much more complex and complete models.

Coupled ocean-atmosphere-ice model (GFDL)(Manabe and Stouffer, 1988)

Interactive atmosphere, ocean, ice/snow modelsGlobal domain, low resolution Land processes representedcloud physics parameterized

Some important extensions to the Stommel model:rotation, full momentum equationmore realistic air-sea fluxeswind-driven circulation

Sea surface temperature and salinity from two model runsthat differ only in their initial conditions

Run A has a much warmer and saltier northern North Atlantic

The surface density in the northern North Atlantic isgreater in run A than it is in run B (salinity is important)

Run A

Run B

A-B

The meridional overturning streamfunction in the Atlantic

Run A has a robust overturning in the North Atlantic, similar inmagnitude to present day conditions

Run B has essentially nooverturning

The northward upper ocean flow isresponsible for the warm, salinesurface conditions in run A. Thisis analogous to the strong, thermalmode of the Stommel model