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Ice/Ocean InteractionPart 2- The Ice/Ocean Interface

1. Heat flux measurements

2. Enthalpy and salt balance at the interface

3. Double diffusion: melting or dissolution?

4. Freezing—is double diffusion important?

AIO Chapter 6

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Ocean Heat Flux

• Relatively small changes in open water fraction when sun angle is high are a major source of variability in the total surface heat balance.

• Contrary to conventional wisdom, the Arctic mixed layer is not an ice bath maintained at freezing temperature– in summer is is typically several tenths of a degree above freezing.

• In the perennial ice pack of the Arctic, transfer of heat from the ocean to the ice occurs mainly in summer, and in aggregate is as important in the overall heat balance as either the net radiative flux, or sensible and latent heat flux at the upper surface.

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DOY 1975

AIDJEX Station Blue Fox, ΔT=T5-Tf(S5)

kelv

ins

Even under compact, multiyear pack ice, the polar mixed layer

spends a good part of the year above freezing.

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NPEO Buoy, Transpolar Drift Stream

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Estimates of interface friction velocity and heat flux from the year-long SHEBA drift in the western Arctic, based on the steady local turbulence model.

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w=w0+w

p

Latent heat source or sink

Turbulent heat flux from ocean

Thermal conduction into iceAdvection into control volume

Advection out of control volume

Ice

water

Thermal Balance at the Ice/Ocean Interface

T0, S

0

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Heat Equation at the Ice/Ocean Interface

• Heat conduction through the ice

• Sensible heat from percolation of fresh water through the ice column

• Latent exchange at the interface

• Turbulent heat flux from (or to) the ocean

Small

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Heat conduction through the ice

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Latent heat exchange at the interface

Turbulent ocean heat flux

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w=w0+w

p

Turbulent salt flux from ocean

Advection into control volume

Advection out of control volume

Ice

Salt Balance at the Ice/Ocean Interface

S0

Sice

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“Kinematic” Interface Heat Equation

Interface Salt Conservation Equation

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Postulate that turbulent heat flux at the interface depends on the following quantities

Use the Pi theorem (dimensional analysis) to show that

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“Two equation” approach

Assumption: δT=Tw-T0 can be approximated by the departure

of mixed layer temperature from freezing, i.e., that interface temperature is approximately the mixed layer freezing temperature. Then the Φ function identified in the dimensional analysis is associated with a bulk Stanton number where the velocity scale is u*0

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Suppose that heat exchange was directly analogous to momentum exchange, hence that the stress can be expressed as proportional to u*0 times the change in

velocity across the boundary layer:

Example problem: If exchange coefficients for heat and momentum are the same, estimate the turbulent heat flux and melt rate for ice drifting with u*0=0.01 m/s in water 1 K above

freezing, assuming that heat conduction in the ice is negligible (QL= 70 K, ρcp=4x106 J kg-1K-1). Under these conditions, how

long would a floe initially 2 m thick last?

u*02 =cmu*0δV

cm = δV /u*0( )−1≈0.1 (very rough MIZ ice)

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Solution:

H f =ρcp w'T ' 0 =4 ×103 W m-2

w 'T '0=cmu*0ΔT =10−3 K m s-1

w0 = w'T ' 0 /QL =10−3 / 70 m s-1

=1.4 ×10−5 m s-1

&h=−w0 / 0.9 ≈140 cm per day

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Yaglom-Kader Re* dependence

From: McPhee, Kottmeier, Morison, 1999, J. Phys. Oceanogr., 29, 1166-1179.

SHEBA

We can measure turbulent stress and heat flux, water temperatureand salinity under sea ice relatively easily. Here are average Stanton numbers from several projects:

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The “two equation” approach then is to express the ocean heat flux in terms of mixed layer properties and interfacial stress, then solve the heat equation for w0 and calculate salinity flux

from w0 and percolation velocity, if present.

However, this is unrealistic for high transfer (melt) rates, because processes at the interface are rate controlled by salt, not heat. They depend on molecular diffusivities in thin layers near the interface, and the ratio νT/νS is about 200 at low

temperatures. Consequently:

w 'S '0=−w(Sice −Sw)

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(1)

(2)

(3)

Heat:

Salt:

Freezing line:

“Three equation” approach

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Combine into quadratic expression for interface salinity

The solution depends on R = αh/αS, which indicates

the strength of the double-diffusive tendency

This equation corresponds to AIO 6.9 except for neglecting a possible “percolation” velocity in the ice column.

mS0

2 + Tw −&q

αhu*0

+αSQL

αh

−mSice

⎛

⎝⎜⎞

⎠⎟S0 −Sice(Tw −

&qαhu*0

)−Sw

αSQL

αh

=0

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In contrast to Stanton number, St*, the interface exchange

coefficients, αh and αS are difficult to measure in the field. There

is theoretical guidance from the Blasius solution for the laminar (nonturbulent) boundary layer, and empirical work from engineering studies of mass transfer over hydraulically rough surfaces:

Yaglom and Kader

Blasius

Owen and Thomson

αh,S ∝ Re*−1/2 (

ν

ν T ,S

)−n

where ν T ,ν S are molecular thermal and haline

diffusivities

ν is molecular kinematic viscosity

and Re* = u*0z0 / ν is a turbulent Reynolds number

2 / 3 ≤n≤0.8

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Sirevaag (2009) has analyzed data from the WARPS project (subset is in WARPS_DATA.mat) and from a combination of heat flux and salinity flux measurements, made direct estimates of αh and αS:

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Indirect Confirmation of Double Diffusion

During the AIDJEX project in the 1970s Arne Hansen had documented the formation and upward migration of thin layers of ice that form at the interface between pools of fresh water that collect in under-ice cavities and the colder seawater.

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During the 1975 AIDJEX Project in the Beaufort Gyre, Arne Hansonmaintained an array of depth gauges at the main station Big Bear. Hereare examples showing a decrease in ice thickness for thick ice, but an increase at several gauges in initially thin ice.

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Thick ice (BB-4 – BB-6) ablated 30-40 cm by the end of melt season. “Falsebottom” gauges showed very little overall ablation during the summer. The box indicates a 10-day period beginning in late July, when false bottoms apparently formed at several sites.

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Assuming a linear temperature gradient in the thin false bottom:

If the upper layer is fresh, at temperature presumably near freezing:

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This modifies the heat equation slightly,but leads to a similar quadratic for S0

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35

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false bottom “true” bottomupward heat flux

down

“water table”

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Summary for melting

Transport of salt across the interface is much slower than heat, and effectively controls the melting rate

The exchange ratios for heat and salt (αh and αS) are difficult to measure, but are constrained by the bulk Stanton number, which is measurable

Collection of fresh water in irregularities in the ice undersurface both protects thin ice from melting and slows the overall heat transfer out of the mixed layer. This retards (provides a negative feedback to) the summer ice-albedo feedback

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Impact of exchange coefficient ratio on freezing with

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Straight congelation growth Growth with frazil accretion

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Hydraulically smooth

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We made measurements as part of the primary experiment from day 67 to 70, then during the UNIS student project a week later. Using the observation that the ice was hydraulically smooth we can estimate the stress from a current meter that recorded continuously at 10 m depth, giving us a 24-day record to provide the momentum forcing for a numerical model.

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During the initial project, the temperature gradient near the base of the ice indicated a conductive heat flux of around 20 W m-2. We used this as a constant flux in the model, shown above for 3 different values of R. The dashed line is the mean growth rate determined by comparing the ice thickness measured at the start and end of the period. The second plot shows the modeled and observed salinity.

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From the numerical model we can estimate the turbulent heat flux 1 m below the interface for the various R values, then compare the model output with measurements made during the two observation period. This provides strong evidence that the double-diffusive effect is very small (R = 1) when ice is freezing.