Impact of resolving the diurnal cycle in an ocean–atmosphere GCM. Part 1: a diurnally forced OGCM

Abstract

The diurnal cycle is a fundamental time scale in the climate system, at which the upper ocean and atmosphere are routinely observed to vary. Current climate models, however, are not configured to resolve the diurnal cycle in the upper ocean or the interaction of the ocean and atmosphere on these time scales. This study examines the diurnal cycle of the tropical upper ocean and its climate impacts. In the present paper, the first of two, a high vertical resolution ocean general circulation model (OGCM), with modified physics, is developed which is able to resolve the diurnal cycle of sea surface temperature (SST) and current variability in the upper ocean. It is then validated against a satellite derived parameterization of diurnal SST variability and in-situ current observations. The model is then used to assess rectification of the intraseasonal SST response to the Madden–Julian oscillation (MJO) by the diurnal cycle of SST. Across the equatorial Indo-Pacific it is found that the diurnal cycle increases the intraseasonal SST response to the MJO by around 20%. In the Pacific, the diurnal cycle also modifies the exchange of momentum between equatorially divergent Ekman currents and the meridionally convergent geostrophic currents beneath, resulting in a 10% increase in the strength of the Ekman cells and equatorial upwelling. How the thermodynamic and dynamical impacts of the diurnal cycle effect the mean state, and variability, of the climate system cannot be fully investigated in the constrained design of ocean-only experiments presented here. The second part of this study, published separately, addresses the climate impacts of the diurnal cycle in the coupled system by coupling the OGCM developed here to an atmosphere general circulation model.

Appendices

Appendix A: Creation of diurnal forcing data

1.1 Reconstruction of diurnal cycle of short wave fluxes

(The IDL code for the following reconstruction of the diurnal cycle of SWF is available from the author on request.)

Daily mean data contains no information about the diurnal cycle of atmospheric convection or cloud cover. Consequently the reconstructed diurnal cycle of SWF is assumed to be simply be a scaling of the idealized top of atmosphere (TOA) incident SWF. If f is the function of the TOA SWF and t is time of day the basic assumption made can be expressed as:

$${{\rm SWR}_{{\rm rec}} {\left(t \right)} = S^{*} f{\left(t \right)}}$$

(2)

where SWF_rec (t ) is reconstructed diurnal cycle of SWF and S ^* is scaling factor. The value of S ^* must be such that:

$${S^{*} {\int\limits_{\rm dawn}^{\rm dusk} {f{\left(t \right)}{\rm d}t =\overline{{\rm SWF}} _{{\rm obs}} {\rm nspd}}}}$$

(3)

where an over-bar represents a daily mean and nspd = 86,400 is the number of seconds per day. Reconstruction of the diurnal cycle of SWF (SWF_rec (t )) will therefore require the determination of f(t), its analytical integral and the limits of this integral so that S ^* can be found by solving Eq. 3. A discrete version of Eq. 2 will then be used to reconstruct the diurnal cycle.

1.2 Diurnal cycle of top of atmosphere short wave fluxes

From some simple geometry the diurnal cycle of the top of atmosphere SWF can be derived as:

$${f{\left({\phi, \theta, \delta, {\rm dss},t} \right)} = S_{0} {\left[{\sin {\left(\phi \right)} \sin {\left({\delta {\left({\rm dss}\right)}} \right)} + \cos {\left(\phi \right)} \cos {\left({\delta{\left({\rm dss} \right)}} \right)} \cos {\left({H_{A}{\left({t,\theta} \right)}} \right)}} \right]}}$$

(4)

where ϕ is latitude, θ longitude, δ declination of the earths orbit which is a function of dss the number of days since the winter solstice (21st December), S ₀ TOA solar constant and H _A , the “hour angle”, is the angle due to the daily rotation of the earth. Here ϕ, θ, δ and H _A are all expressed in radians and that t refers to the time of day, in units of days, in the reference frame of the ERA-40 reanalysis (i.e., UTZ) and so will range from 0 to 1. Consequently:

$${H_{A} = {\left({\theta + 2\pi t - \pi} \right)}}$$

(5)

and

$${\delta = {\left[{{\left({\frac{{- 23.5}}{{360}} 2\pi} \right)} \cos {\left({\frac{{\rm dss}}{{\rm dpy}} 2\pi} \right)}} \right]}}$$

(6)

where dpy is number of days per year. The values in round brackets are to convert declination into radians and express the time of year as a fraction from 0 to 1.

Combining Eqs. 4, 5 and 6 gives a description of the diurnal cycle of the TOA SWF which depends upon longitude, latitude, time of day and time of year. However, this formulation will also produce negative values and so, as will be seen subsequently, care must be taken only to evaluate positive parts of f(t) especially when considering its integral.

1.3 Analytical form and limits of integrated diurnal cycle of short wave fluxes

The integral of f(t) is required to solve Eq. 3. Making the assumption that the declination (δ) is constant over any single (i.e., dss is constant) day and rewriting Eq. 3 for a single day as:

$${f{\left({\phi, \theta, t} \right)} = {\left({A + B \cos{\left({C + D\,t} \right)}} \right)}}$$

(7)

where

$${\begin{aligned} A &= \sin {\left(\phi \right)} \sin{\left(\delta \right)} \\ B &= \cos {\left(\phi \right)} \cos{\left(\delta \right)} \\ C &= {\left({\theta - \pi} \right)} \\ D & =2\pi \\ \end{aligned}}$$

(8)

has the advantage that Eq. 7 can be integrated analytically to give:

$${{\int\limits_{\rm dawn}^{\rm dusk} {f{\left({\phi, \theta, t} \right)}\partial t ={\left[{A\,t + \frac{{B \sin {\left({C + D\,t} \right)}}}{D}}\right]}}}^{{\rm dusk}}_{{\rm dawn}}}.$$

(9)

All that now remains is to determine the limits of Eq. 9 such that Eq. 3 can be solved to find S ^*. By solving Eq. 7 for f(t) = 0

$${\frac{{\cos ^{{- 1}} (\tfrac{{- A}}/{B}) - C}}{D} = t_{x}}$$

(10)

the solution, t _x , gives a time which the SWF will be zero i.e., either dawn or dusk. To determine which of these t _x represents the differential (Eq. 7) with respect to t is taken at t = t _x . A positive result indicating that t _x  = dawn:

$${\begin{aligned} - B\,D \sin {\left({C + D\,t_{x}} \right)} > 0& \Rightarrow \quad {{t_{x} = {\rm dawn}}} \\ - B\, D\, \sin {\left({C +D\,t_{x}} \right)} < 0 &\Rightarrow \quad {{t_{x} = {\rm dusk}}} \\\end{aligned}}.$$

(11)

Now that either dawn or dusk has been found, the other limit can easily be found as the previous assumption that δ is constant over the course of a day produces a diurnal cycle which is symmetrical in time around midday (t _md). Therefore the difference in time between midday and either dawn or dusk allows one to easily find the other limit for Eq. 9 as t _md is solely a function of longitude:

$${t_{{\rm md}} = 0.5 - \frac{\theta}{2}}.$$

(12)

However Eq. 10 only has a solution when there is a distinct day and night, so at high latitudes where there is either 24 h of day or night then |A| > |B| and Eq. 10 has no solution. If this is the case, then A < B implies 24 h of night and so the integral in Eq. 8 is zero. Conversely A > B implies a 24 h day and Eq. 8 should be integrated over the entire day by using the upper and lower limits of 1 and 0, respectively, in Eq. 9.

By using Eqs. 7, 8a–d, 9, 10,11a, b and 12 the limits of Eq. 9 can thus be found and this integral substituted into Eq. 3 to solve for S ^*.

1.4 Reconstruction of discrete diurnal cycle

Analytically, having solved Eq. 3 for S ^* the solution of Eq. 2 with Eq. 7 is trivial. However, to conserve ${\overline{{\rm SWF}} _{{\rm obs}}}$ in a reconstructed diurnal cycle over nts discrete steps requires that:

$${S^{*} {\int\limits_{\rm dawn}^{\rm dusk} {f{\left(t \right)}\partial t =S^{*}}} {\sum\limits_{i = 1}^{\rm nts} {{\left({{\int\limits_{t_{i} -\raise 0.5ex\hbox{$\scriptstyle {\Delta t}$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}^{t_{i}+ \raise0.5ex\hbox{$\scriptstyle {\Delta t}$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}{f{\left(t \right)}\partial t}}} \right)}}} = \overline{{\rm SWF}} _{{\rm obs}} {\rm nspd}}.$$

(13)

However at each time, t _i , Eq. 13 is not necessarily true as:

$${f{\left({t_{i}} \right)} \Delta t \ne {\int\limits_{t_{i} -\raise0.5ex\hbox{$\scriptstyle {\Delta t}$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}^{t_{i}+ \raise0.5ex\hbox{$\scriptstyle {\Delta t}$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}{f{\left(t \right)}}}\partial t}$$

(14)

where nts is required number of time steps per day, Δt length of the time step and t _i time at the centre of each time step. Figure 10 shows graphically the significance of Eq. 14: the value of the analytical solution at t = t _i (red squares) is not the same as the mean value of the analytic solution over a time step (black line). Consequently, the discrete solution for the reconstructed diurnal cycle should be calculated as

$${\rm SWF_{{\rm rec}} {\left({t_{i}} \right)} = \frac{{S^{*} {\int\limits_{t_{i}- \raise0.5ex\hbox{$\scriptstyle {\Delta t}$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}^{t_{i}+ \raise0.5ex\hbox{$\scriptstyle {\Delta t}$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}{f{\left(t \right)}\partial t}}}}{{\Delta t}}}.$$

(15)

(Note that care should be taken not to integrate over any period where f(t) < 0). Using Eq. 15 rather than Eq. 2 will ensure that the following is satisfied and the reconstruction of the diurnal cycle conserves the total flux.

$${\frac{{{\sum\limits_{i = 1}^{\rm nts} {\rm SWF_{{\rm rec}} {\left({t_{i}}\right)}}}}}{{\rm nts}} = \overline{{\rm SWF}} _{{\rm rec}} = \overline{{\rm SWF}}_{{\rm obs}}}.$$

(16)

Fig. 10

Schematic to show graphically the implication of Eqs. 13 and 14; the mean value of the discrete solution is not the same as the analytic solution

Appendix B: List of acronyms

AGCM:

Atmospheric general circulation model

B05:

Bernie et al. 2005

CGCM:

Coupled general circulation model

Dsst:

Diurnal variability of SST

ERA-40:

ECMWF re-analysis

FHDC:

Forced OGCM experiments with high vertical resolution and a diurnal cycle of fluxes

FHDM:

Forced OGCM experiments with high vertical resolution and daily mean fluxes

GCM:

General circulation model

GM04:

Gentemann et al. 2003

HRES:

High vertical Resolution

IMET:

Improved meteorological instrumenT

KPP:

K-profile parameterisation [a first order turbulence closure vertical mixing scheme (Large et al. 1994)]

LOCEAN:

Laboratoire d’Océanographie et de Climatologie par l’Expérimentation et l’Analyse Numérique (formerly LODYC)

LODYC:

Laboratoire d’Océanographie Dynamique et de Climatologie

LOTUS:

Long-term upper ocean study

MB04:

Mellor and Blumberg 2004

MJO:

Madden–Julian oscillation

OGCM:

Ocean general circulation model

ORCA2:

Two degree configuration of the OPA OGCM

OPA:

“Océan Parallélisé”. Ocean model developed at (LODYC) in Paris (Madec et al. 1998)

PF:

PathFinder

PS99:

Price and Sundermeyer 1999

SST:

Sea surface temperature

SWF:

Short wave flux

TKE:

Turbulent kinetic energy [a prognostic kurbulent kinetic energy vertical mixing scheme (Gaspar et al. 1990)]

TOA:

Top of atmosphere

TOGA-COARE:

Tropical ocean–global atmosphere-coupled ocean atmosphere response experiment

TRMM:

Tropical rainfall measurment mission

TVD:

Total variance dissipation